New upper bounds for mixed binary/ternary codes Bart Litjens - PowerPoint PPT Presentation

New upper bounds for mixed binary/ternary codes Bart Litjens Korteweg-de Vries Institute for Mathematics Faculty of Science University of Amsterdam Plze˘ n, October 6th, 2016 Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 1 / 17

Outline Introducing the problem 1 Reductions 2 Results 3 Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 2 / 17

Definitions and notation Let [ m ] := { 0 , ..., m − 1 } , for m ∈ N . Fix n 2 , n 3 , d ∈ Z ≥ 0 . A mixed (binary/ternary) code is a subset of [2] n 2 [3] n 3 . Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 3 / 17

Definitions and notation Let [ m ] := { 0 , ..., m − 1 } , for m ∈ N . Fix n 2 , n 3 , d ∈ Z ≥ 0 . A mixed (binary/ternary) code is a subset of [2] n 2 [3] n 3 . An element of a mixed code is a codeword. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 3 / 17

Definitions and notation Let [ m ] := { 0 , ..., m − 1 } , for m ∈ N . Fix n 2 , n 3 , d ∈ Z ≥ 0 . A mixed (binary/ternary) code is a subset of [2] n 2 [3] n 3 . An element of a mixed code is a codeword. For v , w ∈ [2] n 2 [3] n 3 , the Hamming distance is defined as d H ( v , w ) = |{ i ∈ [ n 2 + n 3 ] | v i � = w i }| . Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 3 / 17

Definitions and notation Let [ m ] := { 0 , ..., m − 1 } , for m ∈ N . Fix n 2 , n 3 , d ∈ Z ≥ 0 . A mixed (binary/ternary) code is a subset of [2] n 2 [3] n 3 . An element of a mixed code is a codeword. For v , w ∈ [2] n 2 [3] n 3 , the Hamming distance is defined as d H ( v , w ) = |{ i ∈ [ n 2 + n 3 ] | v i � = w i }| . The minimum distance d min ( C ) of a code C is the minimum of d H ( v , w ), taken over distinct v , w ∈ C . Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 3 / 17

Definitions and notation Let [ m ] := { 0 , ..., m − 1 } , for m ∈ N . Fix n 2 , n 3 , d ∈ Z ≥ 0 . A mixed (binary/ternary) code is a subset of [2] n 2 [3] n 3 . An element of a mixed code is a codeword. For v , w ∈ [2] n 2 [3] n 3 , the Hamming distance is defined as d H ( v , w ) = |{ i ∈ [ n 2 + n 3 ] | v i � = w i }| . The minimum distance d min ( C ) of a code C is the minimum of d H ( v , w ), taken over distinct v , w ∈ C . Example Let n 2 = n 3 = 2, then d min ( { 01 | 02 , 10 | 12 , 10 | 01 } ) = 2. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 3 / 17

Definitions and notation Let [ m ] := { 0 , ..., m − 1 } , for m ∈ N . Fix n 2 , n 3 , d ∈ Z ≥ 0 . A mixed (binary/ternary) code is a subset of [2] n 2 [3] n 3 . An element of a mixed code is a codeword. For v , w ∈ [2] n 2 [3] n 3 , the Hamming distance is defined as d H ( v , w ) = |{ i ∈ [ n 2 + n 3 ] | v i � = w i }| . The minimum distance d min ( C ) of a code C is the minimum of d H ( v , w ), taken over distinct v , w ∈ C . Example Let n 2 = n 3 = 2, then d min ( { 01 | 02 , 10 | 12 , 10 | 01 } ) = 2. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 4 / 17

The parameter N ( n 2 , n 3 , d ) Definition N ( n 2 , n 3 , d ) := max {| C | | C ⊆ [2] n 2 [3] n 3 , d min ( C ) ≥ d } . Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N ( n 2 , n 3 , d ) Definition N ( n 2 , n 3 , d ) := max {| C | | C ⊆ [2] n 2 [3] n 3 , d min ( C ) ≥ d } . Examples N ( n 2 , n 3 , 1) = 2 n 2 · 3 n 3 = | [2] n 2 [3] n 3 | Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N ( n 2 , n 3 , d ) Definition N ( n 2 , n 3 , d ) := max {| C | | C ⊆ [2] n 2 [3] n 3 , d min ( C ) ≥ d } . Examples N ( n 2 , n 3 , 1) = 2 n 2 · 3 n 3 = | [2] n 2 [3] n 3 | N ( n 2 , n 3 , n 2 + n 3 ) = 2, if n 2 > 0 Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N ( n 2 , n 3 , d ) Definition N ( n 2 , n 3 , d ) := max {| C | | C ⊆ [2] n 2 [3] n 3 , d min ( C ) ≥ d } . Examples N ( n 2 , n 3 , 1) = 2 n 2 · 3 n 3 = | [2] n 2 [3] n 3 | N ( n 2 , n 3 , n 2 + n 3 ) = 2, if n 2 > 0 N (8 , 4 , 3) = 1152 Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N ( n 2 , n 3 , d ) Definition N ( n 2 , n 3 , d ) := max {| C | | C ⊆ [2] n 2 [3] n 3 , d min ( C ) ≥ d } . Examples N ( n 2 , n 3 , 1) = 2 n 2 · 3 n 3 = | [2] n 2 [3] n 3 | N ( n 2 , n 3 , n 2 + n 3 ) = 2, if n 2 > 0 N (8 , 4 , 3) = 1152 Remark Let G = ( V , E ) be the graph with V = [2] n 2 [3] n 3 and E := {{ u , v } | 0 < d H ( u , v ) < d } . Then N ( n 2 , n 3 , d ) = α ( G ) , the stable set number of G. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

Motivation: football pools Source: http://www.uefa.com/uefaeuro/draws/ Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 6 / 17

Motivation: the football pool problem Fix 0 ≤ e ≤ n 2 + n 3 . Suppose n 3 games are played with possible outcome win/draw/loss and n 2 games with possible outcome win/loss. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 7 / 17

Motivation: the football pool problem Fix 0 ≤ e ≤ n 2 + n 3 . Suppose n 3 games are played with possible outcome win/draw/loss and n 2 games with possible outcome win/loss. Covering problem How many forms need to be filled in to make sure that, whatever the outcome, there is at least one form with e good answers? Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 7 / 17

Motivation: the football pool problem Fix 0 ≤ e ≤ n 2 + n 3 . Suppose n 3 games are played with possible outcome win/draw/loss and n 2 games with possible outcome win/loss. Covering problem How many forms need to be filled in to make sure that, whatever the outcome, there is at least one form with e good answers? Packing problem How many forms can be filled in such that, whatever the outcome, there are no two or more forms with more than e good answers? Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 7 / 17

Motivation: the football pool problem Fix 0 ≤ e ≤ n 2 + n 3 . Suppose n 3 games are played with possible outcome win/draw/loss and n 2 games with possible outcome win/loss. Covering problem How many forms need to be filled in to make sure that, whatever the outcome, there is at least one form with e good answers? Packing problem How many forms can be filled in such that, whatever the outcome, there are no two or more forms with more than e good answers? = ⇒ amounts to determining N ( n 2 , n 3 , d ) with d = 2 e + 1. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 7 / 17

Bounds on N ( n 2 , n 3 , d ) Lower bounds: via explicit constructions (Spanish football forum). Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 8 / 17

Bounds on N ( n 2 , n 3 , d ) Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n 2 = 0 or n 3 = 0 case: Delsarte linear programming bound. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 8 / 17

Bounds on N ( n 2 , n 3 , d ) Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n 2 = 0 or n 3 = 0 case: Delsarte linear programming bound. Problem: set of mixed binary/ternary words in general does not form an association scheme with respect to the Hamming distance. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 8 / 17

Bounds on N ( n 2 , n 3 , d ) Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n 2 = 0 or n 3 = 0 case: Delsarte linear programming bound. Problem: set of mixed binary/ternary words in general does not form an association scheme with respect to the Hamming distance. Solution: it has a product scheme structure. Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 8 / 17

Bounds on N ( n 2 , n 3 , d ) Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n 2 = 0 or n 3 = 0 case: Delsarte linear programming bound. Problem: set of mixed binary/ternary words in general does not form an association scheme with respect to the Hamming distance. Solution: it has a product scheme structure. ⇒ Linear programming bound with ≤ ( n 2 + n 3 +1)( n 2 + n 3 +2) = constraints 2 ainen, ¨ (Brouwer, H¨ am¨ al¨ Osterg˚ ard and Sloane, 1998). Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 8 / 17

Semidefinite programming upper bound SDP-bound on N ( n 2 , n 3 , d ) based on triples, k = 3 Let 0 := 0 ... 0 | 0 ... 0 and let C k be the collection of codes C ⊆ [2] n 2 [3] n 3 with | C | ≤ k and 0 ∈ C . Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 9 / 17

Semidefinite programming upper bound SDP-bound on N ( n 2 , n 3 , d ) based on triples, k = 3 Let 0 := 0 ... 0 | 0 ... 0 and let C k be the collection of codes C ⊆ [2] n 2 [3] n 3 with | C | ≤ k and 0 ∈ C . We define � � N 3 ( n 2 , n 3 , d ) := max x ( { 0 , v } ) | x : C 3 → R ≥ 0 with: v ∈ [2] n 2 [3] n 3 (i) x ( { 0 } ) = 1 , (ii) x ( C ) = 0 if d min ( C ) < d , � (iii) M x is positive semidefinite , where M x is the C 2 × C 2 matrix defined by ( M x ) C , C ′ = x ( C ∪ C ′ ) for all C , C ′ ∈ C 2 . Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 9 / 17