Classification of self-orthogonal F q + u F q -codes Classification - PowerPoint PPT Presentation

Self-orthogonal Codes over F q + u F q Rowena Alma Betty, Fidel Nemenzo and Lucky Erap Galvez University of the Philippines-Diliman LAWCI 2018

Classification of self-orthogonal F q + u F q -codes

Classification of self-orthogonal F q + u F q -codes Classification of codes means finding complete set of representatives of equivalence classes .

Classification of self-orthogonal F q + u F q -codes Classification of codes means finding complete set of representatives of equivalence classes . For self-orthogonal codes, establishing the mass formula means finding a formula for | E n | � � | Aut( C ) | = 1 . C [ C ] C : runs through self-orthogonal codes of length n over F q + u F q [ C ]: runs through the equivalence classes of self-orthogonal codes of length n over F q + u F q E n is the full group of transformations allowed in defining the equivalence Aut( C ): automorphism group of C

Survey: Mass formulas for Codes over Some Rings

Survey: Mass formulas for Codes over Some Rings Number of self-dual codes over Z 4 and self-dual codes over F q + u F q , u 2 = 0 (Gaborit)

Survey: Mass formulas for Codes over Some Rings Number of self-dual codes over Z 4 and self-dual codes over F q + u F q , u 2 = 0 (Gaborit) p odd F q + u F q Z p 2 Z 4 s.d. BBN s.d. G G s.o. BM s.o. BM ? G Gaborit BBN Balmaceda-Betty-Nemenzo BM Betty-Munemasa

The ring F q + u F q Let R be the ring F q [ u ] / ( u 2 ) R = = F q + u F q where u commutes with the elements of F q .

The ring F q + u F q Let R be the ring F q [ u ] / ( u 2 ) R = = F q + u F q where u commutes with the elements of F q . The element u is a nilpotent element with nilpotency index 2. This finite chain ring is a local ring with unique maximal ideal ( u ) and F q + u F q / ( u ) = F q is the residue field.

The ring F q + u F q Let x = ( x 1 , x 2 , . . . , x n ) , y = ( y 1 , y 2 , . . . , y n ) ∈ R n . The Euclidean inner product x · y on R n is defined as n � x · y = x i y i . i =1 Let x = a + ub ∈ R and define x = a − ub . The Hermitian inner product x · y on R n is defined as n � x · y = x i y i . i =1

Codes over R Definition A linear code C of length n over R is an R -submodule of the R n module.

Codes over R Definition A linear code C of length n over R is an R -submodule of the R n module. Definition Two codes C and C ′ over R are said to be equivalent if there exists a monomial matrix P such that C ′ = CP = { c · P | c ∈ C }

Codes over R Every code C of length n over R is permutation equivalent to a code with generator matrix � I k 0 � A + uB 0 uD where A, B ∈ M k 0 × ( n − k 0 ) ( F q ) and D ∈ M k 1 × ( n − k 0 ) ( F q ). Such code is said to be of type { k 0 , k 1 } .

Codes over R Every code C of length n over R is permutation equivalent to a code with generator matrix � I k 0 � A + uB 0 uD where A, B ∈ M k 0 × ( n − k 0 ) ( F q ) and D ∈ M k 1 × ( n − k 0 ) ( F q ). Such code is said to be of type { k 0 , k 1 } . Definition Let x = ( x 1 , x 2 , .., x n ) ∈ R n . If C is a code over R , the dual of C , denoted C ⊥ is the set C ⊥ = { x ∈ R n | x · c = 0 ∀ c ∈ C } .

Codes over R Every code C of length n over R is permutation equivalent to a code with generator matrix � I k 0 � A + uB 0 uD where A, B ∈ M k 0 × ( n − k 0 ) ( F q ) and D ∈ M k 1 × ( n − k 0 ) ( F q ). Such code is said to be of type { k 0 , k 1 } . Definition Let x = ( x 1 , x 2 , .., x n ) ∈ R n . If C is a code over R , the dual of C , denoted C ⊥ is the set C ⊥ = { x ∈ R n | x · c = 0 ∀ c ∈ C } . If C ⊆ C ⊥ , C is said to be (Euclidean/Hermitian) self-orthogonal .

Codes over R Every code C of length n over R is permutation equivalent to a code with generator matrix � I k 0 � A + uB 0 uD where A, B ∈ M k 0 × ( n − k 0 ) ( F q ) and D ∈ M k 1 × ( n − k 0 ) ( F q ). Such code is said to be of type { k 0 , k 1 } . Definition Let x = ( x 1 , x 2 , .., x n ) ∈ R n . If C is a code over R , the dual of C , denoted C ⊥ is the set C ⊥ = { x ∈ R n | x · c = 0 ∀ c ∈ C } . If C ⊆ C ⊥ , C is said to be (Euclidean/Hermitian) self-orthogonal . If C = C ⊥ , C is said to be (Euclidean/Hermitian) self-dual .

Codes over R Definition For every code C over R , we associate the codes � v ∈ F n � residue code of C : res( C ) = q | v ∈ C v ∈ F n � � torsion code of C : tor( C ) = q | uv ∈ C

Codes over R Definition For every code C over R , we associate the codes � v ∈ F n � residue code of C : res( C ) = q | v ∈ C v ∈ F n � � torsion code of C : tor( C ) = q | uv ∈ C � I k 0 � A + uB If C has generator matrix , then 0 uD � � 1 res( C ) has generator matrix I k 0 A ;

Codes over R Definition For every code C over R , we associate the codes � v ∈ F n � residue code of C : res( C ) = q | v ∈ C v ∈ F n � � torsion code of C : tor( C ) = q | uv ∈ C � I k 0 � A + uB If C has generator matrix , then 0 uD � � 1 res( C ) has generator matrix I k 0 A ; � I k 0 � A 2 tor( C ) has generator matrix where D is of full 0 D row rank k 1 .

Self-orthogonal Codes over F q + u F q If the code C is self orthogonal over R , then

Self-orthogonal Codes over F q + u F q If the code C is self orthogonal over R , then res( C ) ⊆ tor( C ) ⊆ res( C ) ⊥ .

Self-orthogonal Codes over F q + u F q If the code C is self orthogonal over R , then res( C ) ⊆ tor( C ) ⊆ res( C ) ⊥ . In particular, if C is self-dual, then tor( C ) = res( C ) ⊥ .

Codes over R with prescribed residue and torsion From now on, we assume the following: 1 C 1 is a code of length n over F q with dimension k 0 and generator matrix � � G 1 = I k 0 A 2 C 2 is a code of length n over F q of dimension k 0 + k 1 and generator matrix � I k 0 A � G 2 = 0 D where A, B ∈ M k 0 × ( n − k 0 ) ( F q ) and D ∈ M k 1 × ( n − k 0 ) ( F q ), of full row rank.

Codes over R with prescribed residue and torsion Lemma If C is a code of length n over R with residue code C 1 and torsion code C 2 , then there exists a matrix N ∈ M k 0 × ( n − k 0 ) ( F q ) such that the matrix � I k 0 � A + uN 0 uD is a generator matrix for C . Such matrix N is unique if C is a free code, i.e. k 1 = 0 and C 1 = C 2 .

Codes over R with prescribed residue and torsion Lemma If C is a code of length n over R with residue code C 1 and torsion code C 2 , then there exists a matrix N ∈ M k 0 × ( n − k 0 ) ( F q ) such that the matrix � I k 0 � A + uN 0 uD is a generator matrix for C . Such matrix N is unique if C is a free code, i.e. k 1 = 0 and C 1 = C 2 . Now, we assume that C 1 is self-orthogonal and C 1 ⊆ C 2 ⊆ C ⊥ 1 . Then I k 0 + AA T ≡ 0 ( u ) (1) DA T ≡ 0 ( u ) (2)

Codes over R with prescribed residue and torsion Lemma The number of free Euclidean self-orthogonal codes C over R with given residue (and torsion) code C 1 is q k 0 (2 n − 3 k 0 + ǫ ) / 2 where ǫ = − 1 if char F q � = 2 and ǫ = 1 if char F q = 2 .

Codes over R with prescribed residue and torsion Lemma The number of free Euclidean self-orthogonal codes C over R with given residue (and torsion) code C 1 is q k 0 (2 n − 3 k 0 + ǫ ) / 2 where ǫ = − 1 if char F q � = 2 and ǫ = 1 if char F q = 2 . Proof. C has generator matrix [ I k 0 A + uN ] and since C is self-orthogonal, I k 0 + AA T + u ( AN T + NA T ) 0 ( u 2 ) ≡ AN T + NA T ⇒ ≡ 0 ( u ) by (1)

Codes over R with prescribed residue and torsion Lemma The number of free Euclidean self-orthogonal codes C over R with given residue (and torsion) code C 1 is q k 0 (2 n − 3 k 0 + ǫ ) / 2 where ǫ = − 1 if char F q � = 2 and ǫ = 1 if char F q = 2 . Proof. C has generator matrix [ I k 0 A + uN ] and since C is self-orthogonal, I k 0 + AA T + u ( AN T + NA T ) 0 ( u 2 ) ≡ AN T + NA T ⇒ ≡ 0 ( u ) by (1) Define the map Ψ A : M k 0 × ( n − k 0 ) ( F q ) → M k 0 ( F q ) by Ψ A ( N ) = AN T + NA T .

Codes over R with prescribed residue and torsion Lemma The number of free Euclidean self-orthogonal codes C over R with given residue (and torsion) code C 1 is q k 0 (2 n − 3 k 0 + ǫ ) / 2 where ǫ = − 1 if char F q � = 2 and ǫ = 1 if char F q = 2 . Proof. C has generator matrix [ I k 0 A + uN ] and since C is self-orthogonal, I k 0 + AA T + u ( AN T + NA T ) 0 ( u 2 ) ≡ AN T + NA T ⇒ ≡ 0 ( u ) by (1) Define the map Ψ A : M k 0 × ( n − k 0 ) ( F q ) → M k 0 ( F q ) by Ψ A ( N ) = AN T + NA T . N | AN T + NA T ≡ 0 ( u ) � = | ker Ψ A | � �� �� So,

Ψ A : M k 0 × ( n − k 0 ) ( F q ) → M k 0 ( F q ), Ψ A ( N ) = AN T + NA T � Sym k 0 ( F q ) , char F q � = 2 Moreover, Im(Ψ A ) = . Alt k 0 ( F q ) , char F q = 2 k 0( k 0+1) k 0( k 0 − 1) � = q � � Since � Sym k 0 ( F q ) and | Alt k 0 ( F q ) | = q , 2 2 � q k 0 ( n − k 0 ) − k 0( k 0+1) k 0(2 n − 3 k 0 − 1) = q , char F q � = 2 2 2 | ker Ψ A | = q k 0 ( n − k 0 ) − k 0( k 0 − 1) k 0(2 n − 3 k 0+1) = q , char F q = 2 2 2 �

Codes over R with prescribed residue and torsion Lemma The number of free Hermitian self-orthogonal codes over F q + u F q with given residue (and torsion) code C 1 is q k 0 (2 n − 3 k 0 +1) / 2