Semidefinite programming bounds for codes D. Gijswijt 1 A. Schrijver - PowerPoint PPT Presentation

Introduction Method of Schrijver Computational results Semidefinite programming bounds for codes D. Gijswijt 1 A. Schrijver 2 H. Tanaka 3 1 Department of Operations research Eötvös University, Budapest (Hungary) 2 CWI, Amsterdamy (the Netherlands) 3 Division of Mathematics Tohoku University, Sendai (Japan) Aussois, 2006 D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Method of Schrijver Computational results Outline Introduction 1 Nonbinary codes Method of Delsarte Method of Schrijver 2 A semidefinite program Computational results 3 Tables D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Hamming space Let q = { 0 , 1 , . . . , q − 1 } be an alphabet with q ≥ 3 symbols w ∈ q n is a words of length n . d ( u , v ) := |{ s | u s � = v s }| is the Hamming distance between u , v ∈ q n This makes q n into a metric space, the Hamming space H ( n , q ) . D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Codes A subset C ⊆ q n is a code . with minimum distance min { d ( u , v ) | u , v ∈ C , u � = v } . A q ( n , d ) := maximum cardinality of C ⊆ q n with min. distance d . Goal: find upper bounds for A q ( n , d ) . Note: A q ( n , d ) = α ( G q ( n , d )) with vertices q n and uv edge when d ( u , v ) < d . Bad news: The graph G q ( n , d ) has exponentially many vertices Good news: The Hamming space has a large automorphism group. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Codes A subset C ⊆ q n is a code . with minimum distance min { d ( u , v ) | u , v ∈ C , u � = v } . A q ( n , d ) := maximum cardinality of C ⊆ q n with min. distance d . Goal: find upper bounds for A q ( n , d ) . Note: A q ( n , d ) = α ( G q ( n , d )) with vertices q n and uv edge when d ( u , v ) < d . Bad news: The graph G q ( n , d ) has exponentially many vertices Good news: The Hamming space has a large automorphism group. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Codes A subset C ⊆ q n is a code . with minimum distance min { d ( u , v ) | u , v ∈ C , u � = v } . A q ( n , d ) := maximum cardinality of C ⊆ q n with min. distance d . Goal: find upper bounds for A q ( n , d ) . Note: A q ( n , d ) = α ( G q ( n , d )) with vertices q n and uv edge when d ( u , v ) < d . Bad news: The graph G q ( n , d ) has exponentially many vertices Good news: The Hamming space has a large automorphism group. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Hamming Scheme We need the q n × q n 0–1 matrices A 0 , . . . , A n given by: � 1 if d ( u , v ) = k , ( A k ) u , v := 0 otherwise, for u , v ∈ q n . These matrices span a commutative algebra called the Bose–Mesner algebra of the Hamming scheme . [These are the matrices invariant under permutation of rows and columns by automorphisms] D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Inner distribution The inner distribution ( x 0 , x 1 , . . . , x n ) of a code C is given by x i := ( χ C ) T A i χ C . | C | x i ≥ 0, x 0 = 1, x 1 = x 2 = . . . = x d − 1 = 0, | C | = x 0 + x 1 + . . . + x n , Delsarte constraints... � i x i K j ( i ) ≥ 0 for j = 0 , . . . , n , K j ( i ) = � j r = 0 ( − 1 ) r ( q − 1 ) j − r � i �� n − i � are Krawtchouk r j − r polynomials. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Inner distribution The inner distribution ( x 0 , x 1 , . . . , x n ) of a code C is given by x i := ( χ C ) T A i χ C . | C | x i ≥ 0, x 0 = 1, x 1 = x 2 = . . . = x d − 1 = 0, | C | = x 0 + x 1 + . . . + x n , Delsarte constraints... � i x i K j ( i ) ≥ 0 for j = 0 , . . . , n , K j ( i ) = � j r = 0 ( − 1 ) r ( q − 1 ) j − r � i �� n − i � are Krawtchouk r j − r polynomials. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Inner distribution The inner distribution ( x 0 , x 1 , . . . , x n ) of a code C is given by x i := ( χ C ) T A i χ C . | C | x i ≥ 0, x 0 = 1, x 1 = x 2 = . . . = x d − 1 = 0, | C | = x 0 + x 1 + . . . + x n , Delsarte constraints... � i x i K j ( i ) ≥ 0 for j = 0 , . . . , n , K j ( i ) = � j r = 0 ( − 1 ) r ( q − 1 ) j − r � i �� n − i � are Krawtchouk r j − r polynomials. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Inner distribution The inner distribution ( x 0 , x 1 , . . . , x n ) of a code C is given by x i := ( χ C ) T A i χ C . | C | x i ≥ 0, x 0 = 1, x 1 = x 2 = . . . = x d − 1 = 0, | C | = x 0 + x 1 + . . . + x n , Delsarte constraints... � i x i K j ( i ) ≥ 0 for j = 0 , . . . , n , K j ( i ) = � j r = 0 ( − 1 ) r ( q − 1 ) j − r � i �� n − i � are Krawtchouk r j − r polynomials. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Nonbinary codes Method of Schrijver Method of Delsarte Computational results Delsarte relations The Delsarte inequalities � x i K j ( i ) ≥ 0 , for j = 0 , . . . , n i are equivalent to x i A i · ( v i ) − 1 is positive semidefinite , � i where v i is the number of nonzero entries of A i . D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Method of Schrijver A semidefinite program Computational results Triples of words Consider ordered triples of words ( u , v , w ) ∈ q n × q n × q n . := d ( u , v ) i j := d ( u , w ) # s with u s � = v s and u s � = w s t := p := # s with u s � = v s = w s . We say that d ( u , v , w ) = ( i , j , t , p ) . Observe that d ( v , w ) = i + j − t − p u , v and w all different in t − p positions � n + 4 � tuples ( i , j , t , p ) . 4 D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Method of Schrijver A semidefinite program Computational results Terwilliger algebra For each tuple ( i , j , t , p ) define 0–1 matrix M t , p i , j : � 1 if d ( u , v , w ) = ( i , j , t , p ) , ( M t , p i , j ) u , v := 0 otherwise. The linear span � x t , p i , j M t , p i , j | x t , p A := { i , j ∈ C } i , j , t , p is the Terwilliger algebra . Facts: A is those matrices invariant under permutating rows and columns by symmetries fixing 0. � n + 4 � dim ( A ) = . 4 D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Method of Schrijver A semidefinite program Computational results Counting triples In code C , we count triples of each type. i , j := 1 | C | #triples ( u , v , w ) ∈ C 3 with d ( u , v , w ) = ( i , j , t , p ) . x t , p Observe: x t , p i , j ≥ 0, x 0 , 0 0 , 0 = 1 x t , p i , j = 0 when { i , j , i + j − t − p } ∩ 1 , 2 , . . . , d − 1 � = ∅ i , j = x t ′ , p ′ i ′ , j ′ when t − p = t ′ − p ′ and x t , p { i , j , i + j − t − p } = { i ′ , j ′ , i ′ + j ′ − t ′ − p ′ } . x 0 , 0 0 , 0 + x 0 , 0 1 , 0 + · · · + x 0 , 0 n , 0 = | C | . other. . . D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Method of Schrijver A semidefinite program Computational results Analogue of Delsarte relations Analogous to Delsarte inequalities, the matrices � x t , p i , j ( γ t , p i , j ) − 1 M t , p R ′ := i , j , i , j , t , p i + j − t − p , 0 ) − 1 − x t , p ( x 0 , 0 i + j − y − p , 0 ( γ 0 , 0 i , j ( γ t , p i , j ) − 1 ) M t , p R ′′ � := i , j , i , j , t , p are positive semidefinite. D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Method of Schrijver A semidefinite program Computational results Proof ( χ σ C ) T χ σ C · #autom R ′ = � | C | · q n σ | 0 ∈ σ C ( χ σ C ) T χ σ C · #autom R ′′ = � | C | · q n σ | 0 �∈ σ C � i , j , ( χ σ C ) T χ σ C � = # ( 0 , u , v ) ∈ ( σ C ) 3 with d ( 0 , u , v ) = ( i , j , t , p ) M t , p D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes

Introduction Method of Schrijver A semidefinite program Computational results Proof ( χ σ C ) T χ σ C · #autom R ′ = � | C | · q n σ | 0 ∈ σ C ( χ σ C ) T χ σ C · #autom R ′′ = � | C | · q n σ | 0 �∈ σ C � i , j , ( χ σ C ) T χ σ C � = # ( 0 , u , v ) ∈ ( σ C ) 3 with d ( 0 , u , v ) = ( i , j , t , p ) M t , p D. Gijswijt, A. Schrijver, H. Tanaka SDP bounds for codes