Codes Block designs SGDDs Constructions Self-dual codes from orbit matrices and quotient matrices of combinatorial designs Nina Mostarac (nmavrovic@math.uniri.hr) Dean Crnkovi´ c (deanc@math.uniri.hr) Department of Mathematics, University of Rijeka, Croatia Supported by CSF , Grant 1637 Graphs, groups, and more: celebrating Brian Alspach’s 80th and Dragan Marušiˇ c’s 65th birthdays, Koper, Slovenia June 1, 2018 Nina Mostarac 1/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Content Self-dual codes 1 Block designs 2 Orbit matrices of block designs SGDDs 3 Quotient matrices of SGDDs with the dual property Constructions of self-dual codes 4 Codes from orbit matrices of block designs Codes from symmetric block designs and SGDDs Nina Mostarac 2/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Content Self-dual codes 1 Block designs 2 Orbit matrices of block designs SGDDs 3 Quotient matrices of SGDDs with the dual property Constructions of self-dual codes 4 Codes from orbit matrices of block designs Codes from symmetric block designs and SGDDs Nina Mostarac 3/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Codes Definition 1 Let p be a prime power. A p -ary linear code C of length n and dimension k is a k -dimensional subspace of the vector space ( F p ) n . • Notation: [ n , k ] p code or [ n , k ] code Definition 2 A generating matrix of a linear [ n , k ] code is a k × n matrix whose rows are the basis vectors of the code. Nina Mostarac 4/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Self-dual codes Definition 3 Let C ⊆ F n p be a linear code. Its dual code is the code C ⊥ = { x ∈ F n p | x · c = 0 , ∀ c ∈ C } , where · is the standard inner product. The code C is called self-orthogonal if C ⊆ C ⊥ , and C is called self-dual if C = C ⊥ . Proposition 4 Let G be a generating matrix of a linear [ n , k , d ] code C . 1 C is self-orthogonal ⇔ GG T = 0. 2 C is self-dual ⇔ it is self-orthogonal and k = n 2. Nina Mostarac 5/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Self-dual codes Definition 5 We may use a symmetric nonsingular matrix U over the field F p to define a scalar product �· , ·� U for row vectors in F n p : � a , c � U = aUc T . The U -dual code of a linear code C is the code C U = { a ∈ F n p | � a , c � U = 0 , ∀ c ∈ C } . A code C is called self- U -dual, or self-dual with respect to U , if C = C U . Nina Mostarac 6/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions OM of block designs Content Self-dual codes 1 Block designs 2 Orbit matrices of block designs SGDDs 3 Quotient matrices of SGDDs with the dual property Constructions of self-dual codes 4 Codes from orbit matrices of block designs Codes from symmetric block designs and SGDDs Nina Mostarac 7/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions OM of block designs Block designs Definition 6 A block design or a 2 − ( v , k , λ ) design is a finite incidence structure D = ( P , B , I ) such that |P| = v , each block is incident with exactly k points and each pair of points is incident with exactly λ blocks. If v = b , we say that a block design is symmetric . Nina Mostarac 8/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions OM of block designs Orbit matrices of block designs • Let D = ( P , B , I ) be a 2- ( v , k , λ ) design and let G ≤ Aut ( D ) . • Denote with P 1 , ..., P n G -orbits of points, and with B 1 , ..., B m G -orbits of blocks and let | P i | = ω i , | B j | = Ω j , 1 ≤ i ≤ n , 1 ≤ j ≤ m . • For x ∈ B and Q ∈ P we introduce the notation: � x � = { R ∈ P| ( R , x ) ∈ I } , � Q � = { y ∈ B| ( Q , y ) ∈ I } . • Let Q ∈ P i , x ∈ B j . We will denote: Γ ij = | � Q � ∩ B j | , γ ij = | � x � ∩ P i | . m n � � It holds: Γ ij = r , ∀ i ∈ { 1 , ..., n } , γ ij = k , ∀ j ∈ { 1 , ..., m } . j = 1 i = 1 Definition 7 Matrices S = [Γ ij ] and R = [ γ ij ] are called point and block orbit matrix of the design D induced by the action of the group G . Nina Mostarac 9/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions OM of block designs Lemma 8 Let D = ( P , B , I ) be a block design, G ≤ Aut ( D ) , and let ω i , Ω j , γ ij , Γ ij be defined as before. The following equations hold: a) Ω j γ ij = ω i Γ ij ; m � b) Γ ij γ sj = λω s + δ is · ( r − λ ) , i , s ∈ { 1 , ..., n } . j = 1 Proposition 9 Let D = ( P , B , I ) be a block design, G ≤ Aut ( D ) , and let ω i , Ω j , γ ij , Γ ij be defined as before. The following equations hold: n � γ ij = k ; 1 i = 1 m Ω j � γ ij γ sj = λω s + δ is · ( r − λ ) . 2 ω i j = 1 Nina Mostarac 10/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Quotient matrices of SGDDs with the dual property Content Self-dual codes 1 Block designs 2 Orbit matrices of block designs SGDDs 3 Quotient matrices of SGDDs with the dual property Constructions of self-dual codes 4 Codes from orbit matrices of block designs Codes from symmetric block designs and SGDDs Nina Mostarac 11/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Quotient matrices of SGDDs with the dual property SGDD Definition 10 A (group) divisible design (GDD) with parameters ( v , b , r , k , λ 1 , λ 2 , m , n ) is an incidence structure with v points, b blocks and constant block size k in which every point appears in exactly r blocks and whose point set can be partitioned into m classes of size n , such that: • two points from the same class appear together in exactly λ 1 blocks, • two points from different classes appear together in exactly λ 2 blocks. For the parameters of a GDD it holds: v = mn , bk = vr , ( n − 1 ) λ 1 + n ( m − 1 ) λ 2 = r ( k − 1 ) , rk ≥ v λ 2 . Nina Mostarac 12/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Quotient matrices of SGDDs with the dual property SGDD Definition 11 A GDD is called a symmetric GDD (SGDD) if v = b (or, equivalently, r = k ). It is then denoted by D ( v , k , λ 1 , λ 2 , m , n ) . Definition 12 A SGDD D is said to have the dual property if the dual of D is again a divisible design with the same parameters as D . Nina Mostarac 13/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Quotient matrices of SGDDs with the dual property Quotient matrices of SGDDs with the dual property The point and the block partition from the definition of a SGDD with the dual property give us a canonical partition of the incidence matrix: A 11 · · · A 1 m . . ... . . N = , where A ij ’s are square submatrices of order n . . . A m 1 · · · A mm B 11 · · · B 1 m ⇒ NN T = . . ... . . , B ij = [( k − λ 1 ) I n + ( λ 1 − λ 2 ) J n ] δ ij + λ 2 J n . . B m 1 · · · B mm Nina Mostarac 14/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Quotient matrices of SGDDs with the dual property Quotient matrices of SGDDs with the dual property Remark 1 Each block A ij has constant row (and block) sum. Definition 13 We say that an m × m matrix R = [ r ij ] is a quotient matrix of a SGDD with the dual property if every element r ij is equal to the row sum of the block A ij of the above canonical partition. RR T = ( k 2 − v λ 2 ) I m + n λ 2 J m . It holds: Nina Mostarac 15/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Codes from OM of block designs Codes from symmetric block designs Content Self-dual codes 1 Block designs 2 Orbit matrices of block designs SGDDs 3 Quotient matrices of SGDDs with the dual property Constructions of self-dual codes 4 Codes from orbit matrices of block designs Codes from symmetric block designs and SGDDs Nina Mostarac 16/29 Self-dual codes from combinatorial designs
Codes Block designs SGDDs Constructions Codes from OM of block designs Codes from symmetric block designs Wilson describes the following result of Blokhuis and Calderbank: Theorem 4.1 Let D be a 2 - ( v , k , λ ) design and p an odd prime which exactly divides r − λ (that is p | ( r − λ ) , but p 2 ∤ ( r − λ ) ). Suppose that | S ∩ T | ≡ k ( mod p ) for every two blocks S and T of the design and that v is odd. Then: if k �≡ 0 ( mod p ) , then there exists a self-dual p-ary code of length v + 1 1 with respect to U = diag ( 1 , ..., 1 , − k ) ; if k ≡ 0 ( mod p ) , then there exists a self-dual p-ary code of length v + 1 2 with respect to U ′ = diag ( 1 , ..., 1 , − v ) . Sketch of the proof: Let N be a v × b incidence matrix for D . 0 1 . . . N T , M ′ = . . M = ... N T . 0 1 1 · · · 1 1 Nina Mostarac 17/29 Self-dual codes from combinatorial designs
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