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Weierstrass semigroups at several points, total inflection points on curves and coding theory C cero Carvalho Faculdade de Matem atica Universidade Federal de Uberl andia Algebraic curves over finite fields RICAM - November 1115,

  1. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  2. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  3. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  4. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  5. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  6. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  7. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  8. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  9. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  10. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  11. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  12. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  13. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  14. Weierstrass semigroup at several points We will write n := ( n 1 , . . . , n m ) ∈ N m 0 , e i ∈ N m 0 for the m -tuple that has 1 in the i -th position and 0 in the others, L ( n ) := L ( n 1 P 1 + · · · + n m P m ) and ℓ ( n ) := dim L ( n ). Lemma. The following are equivalent: (i) n ∈ H ; (ii) ℓ ( n ) = ℓ ( n − e i ) + 1 for all i = 1 , . . . , m ; (iii) The linear system | n 1 P 1 + · · · + n m P m | is base-point free. We call N m 0 \ H the set of gaps of H , it is a finite set whose cardinality may vary with P 1 , . . . , P m . For example, if X is a hyperelliptic curve of genus 4, and m = 2 we get: P 2 ✻ P 2 ✻ q q q q q q ❛ q ❛ q ❛ q 26 gaps if P 1 and P 2 are both W. points of X ❛ ❛ ❛ ❛ ❛ q ✲ P 1 ❛ ❛ ❛ q ❛ q ❛ q q ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q 9 gaps if P 1 and P 2 are not W. points of X ✲ P 1 ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q ❛ q ❛ q ❛ q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 3 / 12

  15. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , j � = i , then: (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); (ii) n is a minimal element of the set { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  16. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , j � = i , then: (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); (ii) n is a minimal element of the set { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  17. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , j � = i , then: (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); (ii) n is a minimal element of the set { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  18. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , j � = i , then: (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); (ii) n is a minimal element of the set { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  19. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) P 2 ✻ of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If q q q n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , q ❛ q j � = i , then: q ❛ q ❛ ❛ ❛ q (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); q ❛ q ❛ q ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q (ii) n is a minimal element of the set ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  20. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) P 2 ✻ of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If q q q n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , q ❛ q j � = i , then: q ❛ q ❛ ❛ ❛ q (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); q ❛ q ❛ q ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q (ii) n is a minimal element of the set ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  21. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) P 2 ✻ of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If q q q n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , q ❛ q j � = i , then: q ❛ q ❛ ❛ ❛ q (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); q ❛ q ❛ q ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q (ii) n is a minimal element of the set ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  22. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) P 2 ✻ of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If q q q n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , q ❛ q j � = i , then: q ❛ q ❛ ❛ ❛ q (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); q ❛ q ❛ q ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q (ii) n is a minimal element of the set ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  23. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) P 2 ✻ of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If q q q n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , q ❛ q j � = i , then: q ❛ q ❛ ❛ ❛ q (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); q ❛ q ❛ q ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q (ii) n is a minimal element of the set ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  24. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) P 2 ✻ of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If q q q n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , q ❛ q j � = i , then: q ❛ q ❛ ❛ ❛ q (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); q ❛ q ❛ q ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q (ii) n is a minimal element of the set ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  25. Weierstrass semigroup at several points From now on we assume that #( F ) ≥ m . Properties of H : • For all i = 1 , . . . , m we get that a ∈ H ( P i ) if and only if a . e i ∈ H . • Let ( n 1 , . . . , n m ), ( p 1 , . . . , p m ) ∈ H and set q i := max { n i , p i } , i = 1 , . . . , m . Then ( q 1 , . . . , q m ) ∈ H . Define ( n 1 , . . . , n m ) � ( p 1 , . . . , p m ) if n i ≤ p i ∀ i = 1 , . . . , m . Then � is a partial order in N m 0 . Let i ∈ { 1 , . . . , m } , let n i ∈ N 0 and let n = ( n 1 , . . . , n m ) be a minimal element (w.r.t. � ) P 2 ✻ of the set { ( p 1 , , . . . , p m ) ∈ H | p i = n i } . If q q q n i > 0 and n j > 0 for some j ∈ { 1 , . . . , m } , q ❛ q j � = i , then: q ❛ q ❛ ❛ ❛ q (i) n i e i / ∈ H (hence n i / ∈ H ( P i )); q ❛ q ❛ q ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q (ii) n is a minimal element of the set ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q { ( p 1 , . . . , p m ) ∈ H | p j = n j } , so n j / ∈ H ( P j ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 4 / 12

  26. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / 1 , . . . , m . i n i ≤ γ , where γ is the gonality of (ii) If 1 + � X , then n ∈ G 0 . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  27. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / 1 , . . . , m . i n i ≤ γ , where γ is the gonality of (ii) If 1 + � X , then n ∈ G 0 . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  28. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  29. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  30. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q ∇ 2 ( n ) q q ❛ q q ❛ ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  31. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  32. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q ∇ 2 ( n ) q q ❛ q q ❛ q q ❛ q q q ✲ ∇ 1 ( n ) ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  33. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  34. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  35. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  36. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m P 2 ✻ 0 . q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q 1 , . . . , m . q ❛ q q ❛ q i n i ≤ γ , where γ is the gonality of (ii) If 1 + � ❛ ❛ ❛ q q ❛ q ❛ q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ q q ❛ q ❛ q ❛ q q ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 q ❛ q ❛ q ❛ q ❛ q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  37. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . P 2 P 2 ✻ ✻ Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . q q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q q q q q 1 , . . . , m . ❛ ❛ q q q q ❛ ❛ q q (ii) If 1 + � i n i ≤ γ , where γ is the gonality of ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q ✲ ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 P 1 q ❛ q ❛ q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  38. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . P 2 P 2 ✻ ✻ Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . q q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q q q q q 1 , . . . , m . ❛ ❛ q q q q ❛ ❛ q q (ii) If 1 + � i n i ≤ γ , where γ is the gonality of ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q ✲ ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 P 1 q ❛ q ❛ q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  39. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . P 2 P 2 ✻ ✻ Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . q q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q q q q q 1 , . . . , m . ❛ ❛ q q q q ❛ ❛ q q (ii) If 1 + � i n i ≤ γ , where γ is the gonality of ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q ✲ ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 P 1 q ❛ q ❛ q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  40. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . P 2 P 2 ✻ ✻ Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . q q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q q q q q 1 , . . . , m . ❛ ❛ q q q q ❛ ❛ q q (ii) If 1 + � i n i ≤ γ , where γ is the gonality of ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q ✲ ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 P 1 q ❛ q ❛ q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  41. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . P 2 P 2 ✻ ✻ Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . q q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q q q q q 1 , . . . , m . ❛ ❛ q q q q ❛ ❛ q q (ii) If 1 + � i n i ≤ γ , where γ is the gonality of ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q ✲ ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 P 1 q ❛ q ❛ q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  42. Weierstrass semigroup and AG codes Given n = ( n 1 , . . . , n m ) ∈ N m 0 , define ∇ i ( n ) := { ( p 1 , . . . , p m ) ∈ H | p i = n i and p j ≤ n j ∀ j � = i } Lemma. Let n ∈ N m 0 . The following are equivalent: ∈ H ; (i) n / (ii) ∇ i ( n ) = ∅ for some i ∈ { 1 , . . . , m } . We say that n ∈ N m 0 is a pure gap if ∇ i ( n ) = ∅ for all i ∈ { 1 , . . . , m } . Denote the set of pure gaps by G 0 . P 2 P 2 ✻ ✻ Lemma: Let n = ( n 1 , . . . , n m ) ∈ N m 0 . q q (i) If n ∈ G 0 then n i ∈ H ( P i ) for all i = / q q q q q q 1 , . . . , m . ❛ ❛ q q q q ❛ ❛ q q (ii) If 1 + � i n i ≤ γ , where γ is the gonality of ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q X , then n ∈ G 0 . ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ q ❛ q q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q ✲ ✲ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ P 1 P 1 q ❛ q ❛ q q ❛ ❛ q q ❛ ❛ q q ❛ ❛ q q q q q q q q C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 5 / 12

  43. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  44. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  45. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  46. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  47. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  48. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  49. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  50. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  51. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  52. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  53. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  54. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  55. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  56. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  57. Weierstrass semigroup and AG codes Assume that F is a finite field, let D := Q 1 + · · · + Q n , where Q 1 , . . . , Q n are distinct rational points of X , all distinct from P 1 , . . . , P m , and let G be a divisor with support on P 1 , . . . , P m . Let C Ω ( D , G ) be the algebraic geometry code which is the image of the map ϕ : Ω( G − D ) → F n defined by ϕ ( η ) = ( res Q 1 ( η ) , . . . , res Q n ( η )). Weierstrass semigroup of two rational points in Y 8 + Y = X 9 over F 64 . We know that C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2). Building on previous work by Homma, Kim and Matthews, F. Torres and myself proved the following results. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , and let G = � m i =1 ( n i + p i − 1) P i . Then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m . Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G = � m i =1 ( n i + p i − 1) P i , then C Ω ( D , G ) is an [ n , k , d ]-code, with d ≥ deg( G ) − (2 g − 2) + m + � n i =1 ( p i − n i ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 6 / 12

  58. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  59. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  60. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  61. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  62. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  63. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  64. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  65. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  66. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  67. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  68. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  69. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  70. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  71. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  72. Weierstrass semigroup and AG codes A similar improvement can be made to bounds for the generalized Hamming distance of AG codes. Let r be a positive integer, and C ⊂ F m a linear code. Let U be a subcode of C , the support of U is defined as supp ( U ) := { i | c i � = 0 for some ( c 1 , . . . , c m ) ∈ U } . The r -th generalized Hamming distance of C is defined as d r ( C ) = min { #( supp ( U )) | U is a subcode of C , dim( U ) = r } . Let s be a positive integer and set γ s := { min(deg( A )) | A a divisor with dim L ( A ) = s } . The sequence γ 1 , γ 2 , ... is the gonality sequence of X ; γ 1 = 0 and γ 2 is the gonality of X . This concept was introduced by Yang, Kummar and Stichtenoth in a paper where they proved the following result. Theorem. The r -th generalized Hamming distance of an AG code of length n defined over X satisfies i) d r ( C L ( D , G ) ≥ n − deg( G ) + γ r ; ii) d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2) + γ r . C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 7 / 12

  73. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  74. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  75. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  76. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  77. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  78. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  79. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  80. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  81. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  82. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  83. Weierstrass semigroup and AG codes Using the concept of pure gaps we were able to prove the following result. Theorem. Assume that ( n 1 , . . . , n m ) and ( p 1 , . . . , p m ) are pure gaps of H , with n i ≤ p i for all i = 1 , . . . , m , and that ( q 1 , . . . , q m ) is also a pure gap whenever n i ≤ q i ≤ p i , for all i = 1 , . . . , m . Let G := � m i =1 p i P i . Then: i) d r ( C L ( D , G ) ≥ n − deg( G ) + � m i =1 ( p i − n i ) + m + γ r ; ii) i =1 ( p i − n i )+ m − ( � m d r ( C Ω ( D , G ) ≥ deg( G ) − (2 g − 2)+ γ r + � m i =1 ( p i − n i )+ m ). The last item can lead to an improvement because γ r < γ r +1 for all positive integers r . Thus the existence of pure gaps in H may lead to an improvement of the bounds for the generalized Hamming distances of AG codes. Now we will show that the existence of total inflection points in plane curves determine the existence of pure gaps in certain Weierstrass semigroups. C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 8 / 12

  84. Total inflection points and pure gaps Assume that X is a smooth plane, projective curve, of degree r > 4. We say that P ∈ X is a total inflection point if the tangent line at P intersects X only at P . In a work with T. Kato, we proved the following. Theorem. Let P 1 , P 2 and P 3 be rational, total inflection points of X which do not lie in a line. Then (( r − 4) r , 1 , 1), (1 , ( r − 4) r , 1) and (1 , 1 , ( r − 4) r ) are pure gaps of H ( P 1 , P 2 , P 3 ). Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m j =1 s j , for all i = 1 , . . . , m , and � m j =1 s j ≤ r − 2 − m . Theorem. Let P , P 1 , . . . , P m ∈ X be rational points, with P a total inflection point. Let 0 ≤ i < r − 3 and α 1 , . . . , α m be positive integers such that � m j =1 α j ≤ r − i − 3. Then ( ir + 1 , α 1 , . . . , α m ) is a pure gap of H ( P , P 1 , . . . , P m ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 9 / 12

  85. Total inflection points and pure gaps Assume that X is a smooth plane, projective curve, of degree r > 4. We say that P ∈ X is a total inflection point if the tangent line at P intersects X only at P . In a work with T. Kato, we proved the following. Theorem. Let P 1 , P 2 and P 3 be rational, total inflection points of X which do not lie in a line. Then (( r − 4) r , 1 , 1), (1 , ( r − 4) r , 1) and (1 , 1 , ( r − 4) r ) are pure gaps of H ( P 1 , P 2 , P 3 ). Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m j =1 s j , for all i = 1 , . . . , m , and � m j =1 s j ≤ r − 2 − m . Theorem. Let P , P 1 , . . . , P m ∈ X be rational points, with P a total inflection point. Let 0 ≤ i < r − 3 and α 1 , . . . , α m be positive integers such that � m j =1 α j ≤ r − i − 3. Then ( ir + 1 , α 1 , . . . , α m ) is a pure gap of H ( P , P 1 , . . . , P m ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 9 / 12

  86. Total inflection points and pure gaps Assume that X is a smooth plane, projective curve, of degree r > 4. We say that P ∈ X is a total inflection point if the tangent line at P intersects X only at P . In a work with T. Kato, we proved the following. Theorem. Let P 1 , P 2 and P 3 be rational, total inflection points of X which do not lie in a line. Then (( r − 4) r , 1 , 1), (1 , ( r − 4) r , 1) and (1 , 1 , ( r − 4) r ) are pure gaps of H ( P 1 , P 2 , P 3 ). Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m j =1 s j , for all i = 1 , . . . , m , and � m j =1 s j ≤ r − 2 − m . Theorem. Let P , P 1 , . . . , P m ∈ X be rational points, with P a total inflection point. Let 0 ≤ i < r − 3 and α 1 , . . . , α m be positive integers such that � m j =1 α j ≤ r − i − 3. Then ( ir + 1 , α 1 , . . . , α m ) is a pure gap of H ( P , P 1 , . . . , P m ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 9 / 12

  87. Total inflection points and pure gaps Assume that X is a smooth plane, projective curve, of degree r > 4. We say that P ∈ X is a total inflection point if the tangent line at P intersects X only at P . In a work with T. Kato, we proved the following. Theorem. Let P 1 , P 2 and P 3 be rational, total inflection points of X which do not lie in a line. Then (( r − 4) r , 1 , 1), (1 , ( r − 4) r , 1) and (1 , 1 , ( r − 4) r ) are pure gaps of H ( P 1 , P 2 , P 3 ). Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m j =1 s j , for all i = 1 , . . . , m , and � m j =1 s j ≤ r − 2 − m . Theorem. Let P , P 1 , . . . , P m ∈ X be rational points, with P a total inflection point. Let 0 ≤ i < r − 3 and α 1 , . . . , α m be positive integers such that � m j =1 α j ≤ r − i − 3. Then ( ir + 1 , α 1 , . . . , α m ) is a pure gap of H ( P , P 1 , . . . , P m ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 9 / 12

  88. Total inflection points and pure gaps Assume that X is a smooth plane, projective curve, of degree r > 4. We say that P ∈ X is a total inflection point if the tangent line at P intersects X only at P . In a work with T. Kato, we proved the following. Theorem. Let P 1 , P 2 and P 3 be rational, total inflection points of X which do not lie in a line. Then (( r − 4) r , 1 , 1), (1 , ( r − 4) r , 1) and (1 , 1 , ( r − 4) r ) are pure gaps of H ( P 1 , P 2 , P 3 ). Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m j =1 s j , for all i = 1 , . . . , m , and � m j =1 s j ≤ r − 2 − m . Theorem. Let P , P 1 , . . . , P m ∈ X be rational points, with P a total inflection point. Let 0 ≤ i < r − 3 and α 1 , . . . , α m be positive integers such that � m j =1 α j ≤ r − i − 3. Then ( ir + 1 , α 1 , . . . , α m ) is a pure gap of H ( P , P 1 , . . . , P m ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 9 / 12

  89. Total inflection points and pure gaps Assume that X is a smooth plane, projective curve, of degree r > 4. We say that P ∈ X is a total inflection point if the tangent line at P intersects X only at P . In a work with T. Kato, we proved the following. Theorem. Let P 1 , P 2 and P 3 be rational, total inflection points of X which do not lie in a line. Then (( r − 4) r , 1 , 1), (1 , ( r − 4) r , 1) and (1 , 1 , ( r − 4) r ) are pure gaps of H ( P 1 , P 2 , P 3 ). Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m j =1 s j , for all i = 1 , . . . , m , and � m j =1 s j ≤ r − 2 − m . Theorem. Let P , P 1 , . . . , P m ∈ X be rational points, with P a total inflection point. Let 0 ≤ i < r − 3 and α 1 , . . . , α m be positive integers such that � m j =1 α j ≤ r − i − 3. Then ( ir + 1 , α 1 , . . . , α m ) is a pure gap of H ( P , P 1 , . . . , P m ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 9 / 12

  90. Total inflection points and pure gaps Assume that X is a smooth plane, projective curve, of degree r > 4. We say that P ∈ X is a total inflection point if the tangent line at P intersects X only at P . In a work with T. Kato, we proved the following. Theorem. Let P 1 , P 2 and P 3 be rational, total inflection points of X which do not lie in a line. Then (( r − 4) r , 1 , 1), (1 , ( r − 4) r , 1) and (1 , 1 , ( r − 4) r ) are pure gaps of H ( P 1 , P 2 , P 3 ). Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m j =1 s j , for all i = 1 , . . . , m , and � m j =1 s j ≤ r − 2 − m . Theorem. Let P , P 1 , . . . , P m ∈ X be rational points, with P a total inflection point. Let 0 ≤ i < r − 3 and α 1 , . . . , α m be positive integers such that � m j =1 α j ≤ r − i − 3. Then ( ir + 1 , α 1 , . . . , α m ) is a pure gap of H ( P , P 1 , . . . , P m ). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 9 / 12

  91. Total inflection points and pure gaps Theorem. Let P 1 , . . . , P m be total inflection points on X . Then ( s 1 r + α 1 , . . . , s m r + α m ) is a pure gap of H ( P 1 , . . . , P m ), whenever s i , α i are integers such that s i ≥ 0, 1 ≤ α i ≤ r − 1 − i − � m i =1 s i , for all i = 1 , . . . , m , and � m i =1 s i ≤ r − 2 − m . Application. Take X ⊂ P 2 ( K ) the Hermitian curve of degree q + 1 defined over F = GF ( q 2 ). Let s and m be positive integers such that s + m ≤ q − 1; let P 1 , . . . , P m be distinct rational points of X . Take s 1 = s , s 2 = · · · = s m = 0, from the above theorem we get that ( sr + α 1 , α 2 , . . . , α m ) is a pure gap at H ( P 1 , . . . , P m ) whenever 1 ≤ α i ≤ q − i − s ( i = 1 , . . . , m ) (i.e. only have pure gaps “between” ( sr + 1 , 1 , . . . , 1) and ( sr + q − 1 − s , q − 2 − s , . . . , q − m − s )). Let G = (2 sr + q − 1 − s ) P 1 + � m i =2 ( q − i − s ) P i and let D be the sum of the other q 3 + 1 − m rational points of X . From the work together with F. Torres we know that C Ω ( D , G ) is an [ q 3 + 1 − m , k , d ] code with d ≥ 2 s ( q + 1) + m (2 q − 2 s − 1) − m 2 − ( q + 1)( q − 2). C´ ıcero Carvalho (UFU) Inflection points on curves and coding theory 10 / 12

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