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

1

Introducing the problem

2

Reductions

3

Results

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 n2, n3, d ∈ Z≥0. A mixed (binary/ternary) code is a subset of [2]n2[3]n3.

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 n2, n3, d ∈ Z≥0. A mixed (binary/ternary) code is a subset of [2]n2[3]n3. 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 n2, n3, d ∈ Z≥0. A mixed (binary/ternary) code is a subset of [2]n2[3]n3. An element of a mixed code is a codeword. For v, w ∈ [2]n2[3]n3, the Hamming distance is defined as dH(v, w) = |{i ∈ [n2 + n3] | vi = wi}|.

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 n2, n3, d ∈ Z≥0. A mixed (binary/ternary) code is a subset of [2]n2[3]n3. An element of a mixed code is a codeword. For v, w ∈ [2]n2[3]n3, the Hamming distance is defined as dH(v, w) = |{i ∈ [n2 + n3] | vi = wi}|. The minimum distance dmin(C) of a code C is the minimum of dH(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 n2, n3, d ∈ Z≥0. A mixed (binary/ternary) code is a subset of [2]n2[3]n3. An element of a mixed code is a codeword. For v, w ∈ [2]n2[3]n3, the Hamming distance is defined as dH(v, w) = |{i ∈ [n2 + n3] | vi = wi}|. The minimum distance dmin(C) of a code C is the minimum of dH(v, w), taken over distinct v, w ∈ C.

Example

Let n2 = n3 = 2, then dmin({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 n2, n3, d ∈ Z≥0. A mixed (binary/ternary) code is a subset of [2]n2[3]n3. An element of a mixed code is a codeword. For v, w ∈ [2]n2[3]n3, the Hamming distance is defined as dH(v, w) = |{i ∈ [n2 + n3] | vi = wi}|. The minimum distance dmin(C) of a code C is the minimum of dH(v, w), taken over distinct v, w ∈ C.

Example

Let n2 = n3 = 2, then dmin({01|02, 10|12, 10|01}) = 2.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 4 / 17

The parameter N(n2, n3, d)

Definition

N(n2, n3, d) := max{|C| | C ⊆ [2]n2[3]n3, dmin(C) ≥ d}.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N(n2, n3, d)

Definition

N(n2, n3, d) := max{|C| | C ⊆ [2]n2[3]n3, dmin(C) ≥ d}.

Examples

N(n2, n3, 1) = 2n2 · 3n3 = |[2]n2[3]n3|

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N(n2, n3, d)

Definition

N(n2, n3, d) := max{|C| | C ⊆ [2]n2[3]n3, dmin(C) ≥ d}.

Examples

N(n2, n3, 1) = 2n2 · 3n3 = |[2]n2[3]n3| N(n2, n3, n2 + n3) = 2, if n2 > 0

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N(n2, n3, d)

Definition

N(n2, n3, d) := max{|C| | C ⊆ [2]n2[3]n3, dmin(C) ≥ d}.

Examples

N(n2, n3, 1) = 2n2 · 3n3 = |[2]n2[3]n3| N(n2, n3, n2 + n3) = 2, if n2 > 0 N(8, 4, 3) = 1152

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 5 / 17

The parameter N(n2, n3, d)

Definition

N(n2, n3, d) := max{|C| | C ⊆ [2]n2[3]n3, dmin(C) ≥ d}.

Examples

N(n2, n3, 1) = 2n2 · 3n3 = |[2]n2[3]n3| N(n2, n3, n2 + n3) = 2, if n2 > 0 N(8, 4, 3) = 1152

Remark

Let G = (V , E) be the graph with V = [2]n2[3]n3 and E := {{u, v} | 0 < dH(u, v) < d}. Then N(n2, n3, 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 ≤ n2 + n3. Suppose n3 games are played with possible outcome win/draw/loss and n2 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 ≤ n2 + n3. Suppose n3 games are played with possible outcome win/draw/loss and n2 games with possible outcome win/loss.

Covering problem

How many forms need to be filled in to make sure that, whatever the

• utcome, 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 ≤ n2 + n3. Suppose n3 games are played with possible outcome win/draw/loss and n2 games with possible outcome win/loss.

Covering problem

How many forms need to be filled in to make sure that, whatever the

• utcome, 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 ≤ n2 + n3. Suppose n3 games are played with possible outcome win/draw/loss and n2 games with possible outcome win/loss.

Covering problem

How many forms need to be filled in to make sure that, whatever the

• utcome, 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(n2, n3, d) with d = 2e + 1.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 7 / 17

Bounds on N(n2, n3, 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(n2, n3, d)

Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n2 = 0 or n3 = 0 case: Delsarte linear programming bound.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 8 / 17

Bounds on N(n2, n3, d)

Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n2 = 0 or n3 = 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(n2, n3, d)

Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n2 = 0 or n3 = 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(n2, n3, d)

Lower bounds: via explicit constructions (Spanish football forum). Upper bound for n2 = 0 or n3 = 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 ≤ (n2+n3+1)(n2+n3+2)

2

constraints (Brouwer, H¨ am¨ al¨ ainen, ¨ 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(n2, n3, d) based on triples, k = 3

Let 0 := 0...0|0...0 and let Ck be the collection of codes C ⊆ [2]n2[3]n3 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(n2, n3, d) based on triples, k = 3

Let 0 := 0...0|0...0 and let Ck be the collection of codes C ⊆ [2]n2[3]n3 with |C| ≤ k and 0 ∈ C. We define N3(n2, n3, d) := max

• v∈[2]n2[3]n3

x({0, v}) | x : C3 → R≥0 with: (i) x({0}) = 1, (ii) x(C) = 0 if dmin(C) < d, (iii) Mx is positive semidefinite

• ,

where Mx is the C2 × C2 matrix defined by (Mx)C,C ′ = x(C ∪ C ′) for all C, C ′ ∈ C2.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 9 / 17

Upper bound

Theorem

N(n2, n3, d) ≤ N3(n2, n3, d).

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 10 / 17

Upper bound

Theorem

N(n2, n3, d) ≤ N3(n2, n3, d).

Proof.

Let S ⊂ [2]n2[3]n3 be a code with dmin(S) ≥ d and |S| = N(n2, n3, d).

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 10 / 17

Upper bound

Theorem

N(n2, n3, d) ≤ N3(n2, n3, d).

Proof.

Let S ⊂ [2]n2[3]n3 be a code with dmin(S) ≥ d and |S| = N(n2, n3, d). We may assume that 0 ∈ S.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 10 / 17

Upper bound

Theorem

N(n2, n3, d) ≤ N3(n2, n3, d).

Proof.

Let S ⊂ [2]n2[3]n3 be a code with dmin(S) ≥ d and |S| = N(n2, n3, d). We may assume that 0 ∈ S. Define x : C3 → R≥0, x(C) =

• 1 if C ⊂ S

0 otherwise

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 10 / 17

Upper bound

Theorem

N(n2, n3, d) ≤ N3(n2, n3, d).

Proof.

Let S ⊂ [2]n2[3]n3 be a code with dmin(S) ≥ d and |S| = N(n2, n3, d). We may assume that 0 ∈ S. Define x : C3 → R≥0, x(C) =

• 1 if C ⊂ S

0 otherwise Conditions (i) and (ii) are satisfied.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 10 / 17

Upper bound

Theorem

N(n2, n3, d) ≤ N3(n2, n3, d).

Proof.

Let S ⊂ [2]n2[3]n3 be a code with dmin(S) ≥ d and |S| = N(n2, n3, d). We may assume that 0 ∈ S. Define x : C3 → R≥0, x(C) =

• 1 if C ⊂ S

0 otherwise Conditions (i) and (ii) are satisfied. Since (Mx)C,C ′ = x(C)x(C ′) for all C, C ′ ∈ C2, condition (iii) is also satisfied.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 10 / 17

Upper bound

Theorem

N(n2, n3, d) ≤ N3(n2, n3, d).

Proof.

Let S ⊂ [2]n2[3]n3 be a code with dmin(S) ≥ d and |S| = N(n2, n3, d). We may assume that 0 ∈ S. Define x : C3 → R≥0, x(C) =

• 1 if C ⊂ S

0 otherwise Conditions (i) and (ii) are satisfied. Since (Mx)C,C ′ = x(C)x(C ′) for all C, C ′ ∈ C2, condition (iii) is also satisfied. Now

• v∈[2]n2[3]n3

x({0, v}) = |S| = N(n2, n3, d).

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 10 / 17

Outline

1

Introducing the problem

2

Reductions

3

Results

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 11 / 17

The isometry group

The group of Hamming distance preserving permutations of [2]n2[3]n3 is (Sn2

2 ⋊ Sn2) × (Sn3 3 ⋊ Sn3).

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 12 / 17

The isometry group

The group of Hamming distance preserving permutations of [2]n2[3]n3 is (Sn2

2 ⋊ Sn2) × (Sn3 3 ⋊ Sn3).

Let G be the subgroup of isometries leaving 0 invariant, then G = Sn2 × (Sn3

2 ⋊ Sn3).

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 12 / 17

The isometry group

The group of Hamming distance preserving permutations of [2]n2[3]n3 is (Sn2

2 ⋊ Sn2) × (Sn3 3 ⋊ Sn3).

Let G be the subgroup of isometries leaving 0 invariant, then G = Sn2 × (Sn3

2 ⋊ Sn3).

The group G acts on C3, preserving minimum distances.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 12 / 17

The isometry group

The group of Hamming distance preserving permutations of [2]n2[3]n3 is (Sn2

2 ⋊ Sn2) × (Sn3 3 ⋊ Sn3).

Let G be the subgroup of isometries leaving 0 invariant, then G = Sn2 × (Sn3

2 ⋊ Sn3).

The group G acts on C3, preserving minimum distances. If x : C3 → R≥0 is an optimal solution, then also xπ given by xπ(C) := x(π ◦ C) is optimal for all π ∈ G.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 12 / 17

The isometry group

The group of Hamming distance preserving permutations of [2]n2[3]n3 is (Sn2

2 ⋊ Sn2) × (Sn3 3 ⋊ Sn3).

Let G be the subgroup of isometries leaving 0 invariant, then G = Sn2 × (Sn3

2 ⋊ Sn3).

The group G acts on C3, preserving minimum distances. If x : C3 → R≥0 is an optimal solution, then also xπ given by xπ(C) := x(π ◦ C) is optimal for all π ∈ G. Then y := (1/|G|)

π∈G xπ is a G-invariant optimal solution.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 12 / 17

G-orbits of C3

Let Ω be the set of G-orbits of C3.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 13 / 17

G-orbits of C3

Let Ω be the set of G-orbits of C3.

Fact

|Ω| bounded by a polynomial in n2 and n3.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 13 / 17

G-orbits of C3

Let Ω be the set of G-orbits of C3.

Fact

|Ω| bounded by a polynomial in n2 and n3. Replace variable x(C), with C ∈ C3, by y(w), with w ∈ Ω the orbit containing C.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 13 / 17

G-orbits of C3

Let Ω be the set of G-orbits of C3.

Fact

|Ω| bounded by a polynomial in n2 and n3. Replace variable x(C), with C ∈ C3, by y(w), with w ∈ Ω the orbit containing C. Get a matrix My, that is invariant under the action of G on rows and columns.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 13 / 17

G-orbits of C3

Let Ω be the set of G-orbits of C3.

Fact

|Ω| bounded by a polynomial in n2 and n3. Replace variable x(C), with C ∈ C3, by y(w), with w ∈ Ω the orbit containing C. Get a matrix My, that is invariant under the action of G on rows and columns. = ⇒ My ∈ EndG(RC2)

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 13 / 17

An explicit block diagonalization

Representation theory of the group G yields a matrix U, independent of y, such that

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 14 / 17

An explicit block diagonalization

Representation theory of the group G yields a matrix U, independent of y, such that

Theorem (Maschke’s theorem + Schur’s lemma)

EndG(RC2) ∼ − →

i Rmi×mi (as linear spaces), via A → UtAU.

Moreover, A is positive semidefinite if and only if each of the blocks of UtAU is.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 14 / 17

An explicit block diagonalization

Representation theory of the group G yields a matrix U, independent of y, such that

Theorem (Maschke’s theorem + Schur’s lemma)

EndG(RC2) ∼ − →

i Rmi×mi (as linear spaces), via A → UtAU.

Moreover, A is positive semidefinite if and only if each of the blocks of UtAU is. Blocks and coefficients of the constraint matrices can be obtained from the n2 = 0 and n3 = 0 case.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 14 / 17

Outline

1

Introducing the problem

2

Reductions

3

Results

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 15 / 17

Results

Results (L., 2016)

In total 135 improved upper bounds were found: 131 from the SDP with k = 3, one new bound from the SDP with k = 4 and three implicit improvements.

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 16 / 17

A selection of the results

Table: A part of the table with best known bounds on N(n2, n3, 4). The improved bounds are boldfaced.

n2\n3 2 3 4 5 6 2 2 3 8 22 51-61 3 3 6 15 36-43 92-117 4 6 11 28-30 62-83 158-228 5 8 20 50-59 114-160 288-436 6 16 34-40 96-114 216-308 576-825 7 36-30 64-80 192-220 408-585 1152-1576 8 50-59 128-153 384-407 768-1103 2304-3027 9 96-108 256-288 548-771 1536-2105 10 192-212 420-548 1050-1480 11 384 784-1032

Bart Litjens Semidefinite code bounds AGT Plze˘ n, Oct. 6th, 2016 17 / 17