New upper bounds for nonbinary codes based on semidefinite - PowerPoint PPT Presentation

New upper bounds for nonbinary codes based on semidefinite programming and parity Sven Polak Partly based on joint work with Bart Litjens and Lex Schrijver Korteweg-de Vries Institute for Mathematics Faculty of Science University of Amsterdam Plzeˇ n, October 6th, 2016 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Definitions and notation Fix q , n , d ∈ N with q ≥ 2. Define [ q ] := { 0 , . . . , q − 1 } . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Definitions and notation Fix q , n , d ∈ N with q ≥ 2. Define [ q ] := { 0 , . . . , q − 1 } . A word is an element of [ q ] n and a code is a subset of [ q ] n . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Definitions and notation Fix q , n , d ∈ N with q ≥ 2. Define [ q ] := { 0 , . . . , q − 1 } . A word is an element of [ q ] n and a code is a subset of [ q ] n . The Hamming distance between two words u , v ∈ [ q ] n is d H ( u , v ) := |{ i : u i � = v i }| . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Definitions and notation Fix q , n , d ∈ N with q ≥ 2. Define [ q ] := { 0 , . . . , q − 1 } . A word is an element of [ q ] n and a code is a subset of [ q ] n . The Hamming distance between two words u , v ∈ [ q ] n is d H ( u , v ) := |{ i : u i � = v i }| . The minimum distance d min ( C ) of a code C ⊆ [ q ] n is the minimum of d H ( u , v ) over all distinct u , v ∈ C . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Definitions and notation Fix q , n , d ∈ N with q ≥ 2. Define [ q ] := { 0 , . . . , q − 1 } . A word is an element of [ q ] n and a code is a subset of [ q ] n . The Hamming distance between two words u , v ∈ [ q ] n is d H ( u , v ) := |{ i : u i � = v i }| . The minimum distance d min ( C ) of a code C ⊆ [ q ] n is the minimum of d H ( u , v ) over all distinct u , v ∈ C . Example (i) d min ( { 1112 , 2111 , 3134 } ) = 2, Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Definitions and notation Fix q , n , d ∈ N with q ≥ 2. Define [ q ] := { 0 , . . . , q − 1 } . A word is an element of [ q ] n and a code is a subset of [ q ] n . The Hamming distance between two words u , v ∈ [ q ] n is d H ( u , v ) := |{ i : u i � = v i }| . The minimum distance d min ( C ) of a code C ⊆ [ q ] n is the minimum of d H ( u , v ) over all distinct u , v ∈ C . Example (i) d min ( { 1112 , 2111 , 3134 } ) = 2, Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Examples A q ( n , 1) = q n . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Examples A q ( n , 1) = q n . A 5 (7 , 6) = 15. Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Examples A q ( n , 1) = q n . A 5 (7 , 6) = 15. (i) Tables with bounds on A q ( n , d ) on the website of Andries Brouwer. Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Examples A q ( n , 1) = q n . A 5 (7 , 6) = 15. (i) Tables with bounds on A q ( n , d ) on the website of Andries Brouwer. (ii) Interesting parameter in cryptography: a code C ⊆ [ q ] n with d min ( C ) = 2 e + 1 is e - error correcting . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) – II Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) – II Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Remark Let G = ( V , E ) be the graph with V = [ q ] n and E := {{ u , v } | 0 < d H ( u , v ) < d } . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) – II Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Remark Let G = ( V , E ) be the graph with V = [ q ] n and E := {{ u , v } | 0 < d H ( u , v ) < d } . Then A q ( n , d ) = α ( G ), the stable set number of G . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) – II Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Remark Let G = ( V , E ) be the graph with V = [ q ] n and E := {{ u , v } | 0 < d H ( u , v ) < d } . Then A q ( n , d ) = α ( G ), the stable set number of G . 000 001 101 010 n = 3 , d = 2 110 011 111 100 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) – II Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Remark Let G = ( V , E ) be the graph with V = [ q ] n and E := {{ u , v } | 0 < d H ( u , v ) < d } . Then A q ( n , d ) = α ( G ), the stable set number of G . 000 001 101 010 n = 3 , d = 2 110 011 111 100 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) – II Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Remark Let G = ( V , E ) be the graph with V = [ q ] n and E := {{ u , v } | 0 < d H ( u , v ) < d } . Then A q ( n , d ) = α ( G ), the stable set number of G . 000 001 101 010 n = 3 , d = 2 110 011 111 100 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The parameter A q ( n , d ) – II Definition A q ( n , d ) := max {| C | | C ⊆ [ q ] n , d min ( C ) ≥ d } . Remark Let G = ( V , E ) be the graph with V = [ q ] n and E := {{ u , v } | 0 < d H ( u , v ) < d } . Then A q ( n , d ) = α ( G ), the stable set number of G . 000 001 101 010 n = 3 , d = 2 110 011 111 100 Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Delsarte bound Notation Let C k be the collection of codes C ⊆ [ q ] n with | C | ≤ k . Given x : C 2 → R ≥ 0 , define the C 1 × C 1 -matrix M x by ( M x ) C , C ′ = x ( C ∪ C ′ ) . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

Delsarte bound Notation Let C k be the collection of codes C ⊆ [ q ] n with | C | ≤ k . Given x : C 2 → R ≥ 0 , define the C 1 × C 1 -matrix M x by ( M x ) C , C ′ = x ( C ∪ C ′ ) . It can be proven that the Delsarte bound equals � � θ q ( n , d ) := max x ( { v } ) | x : C 2 → R ≥ 0 with: v ∈ [ q ] n (i) x ( ∅ ) = 1 , (ii) x ( C ) = 0 if d min ( C ) < d , � (iii) M x is positive semidefinite . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The quadruple bound Notation Let C k be the collection of codes C ⊆ [ q ] n with | C | ≤ k . Given x : C 4 → R ≥ 0 , define the C 2 × C 2 -matrix M x by ( M x ) C , C ′ = x ( C ∪ C ′ ) . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The quadruple bound Notation Let C k be the collection of codes C ⊆ [ q ] n with | C | ≤ k . Given x : C 4 → R ≥ 0 , define the C 2 × C 2 -matrix M x by ( M x ) C , C ′ = x ( C ∪ C ′ ) . Now we define Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The quadruple bound Notation Let C k be the collection of codes C ⊆ [ q ] n with | C | ≤ k . Given x : C 4 → R ≥ 0 , define the C 2 × C 2 -matrix M x by ( M x ) C , C ′ = x ( C ∪ C ′ ) . Now we define � � x ( { v } ) | x : C 4 → R ≥ 0 with: B q ( n , d ) := max v ∈ [ q ] n (i) x ( ∅ ) = 1 , (ii) x ( C ) = 0 if d min ( C ) < d , � (iii) M x is positive semidefinite . Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The quadruple bound � � x ( { v } ) | x : C 4 → R ≥ 0 with: B q ( n , d ) := max v ∈ [ q ] n (i) x ( ∅ ) = 1 , (ii) x ( C ) = 0 if d min ( C ) < d , � (iii) M x is positive semidefinite . Proposition. A q ( n , d ) ≤ B q ( n , d ) Proof. Let C ⊆ [ q ] n be a code of minimum distance at least d Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The quadruple bound � � x ( { v } ) | x : C 4 → R ≥ 0 with: B q ( n , d ) := max v ∈ [ q ] n (i) x ( ∅ ) = 1 , (ii) x ( C ) = 0 if d min ( C ) < d , � (iii) M x is positive semidefinite . Proposition. A q ( n , d ) ≤ B q ( n , d ) Proof. Let C ⊆ [ q ] n be a code of minimum distance at least d and maximum size. Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The quadruple bound � � x ( { v } ) | x : C 4 → R ≥ 0 with: B q ( n , d ) := max v ∈ [ q ] n (i) x ( ∅ ) = 1 , (ii) x ( C ) = 0 if d min ( C ) < d , � (iii) M x is positive semidefinite . Proposition. A q ( n , d ) ≤ B q ( n , d ) Proof. Let C ⊆ [ q ] n be a code of minimum distance at least d and maximum size. Define x by x ( S ) = 1 if S ⊆ C and x ( S ) = 0 else. Sven Polak Code bounds Plzeˇ n, October 6th, 2016

The quadruple bound � � x ( { v } ) | x : C 4 → R ≥ 0 with: B q ( n , d ) := max v ∈ [ q ] n (i) x ( ∅ ) = 1 , (ii) x ( C ) = 0 if d min ( C ) < d , � (iii) M x is positive semidefinite . Proposition. A q ( n , d ) ≤ B q ( n , d ) Proof. Let C ⊆ [ q ] n be a code of minimum distance at least d and maximum size. Define x by x ( S ) = 1 if S ⊆ C and x ( S ) = 0 else. Then x is feasible. Sven Polak Code bounds Plzeˇ n, October 6th, 2016