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

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

May 21, 2023 284 likes •536 views

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 Etvs University, Budapest (Hungary) 2 CWI, Amsterdamy (the

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SLIDE 1

Introduction Method of Schrijver Computational results

Semidefinite programming bounds for codes

  • D. Gijswijt1
  • A. Schrijver2
  • H. Tanaka3

1Department of Operations research

Eötvös University, Budapest (Hungary)

2CWI, Amsterdamy (the Netherlands) 3Division of Mathematics

Tohoku University, Sendai (Japan)

Aussois, 2006

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 2

Introduction Method of Schrijver Computational results

Outline

1

Introduction Nonbinary codes Method of Delsarte

2

Method of Schrijver A semidefinite program

3

Computational results Tables

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 3

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Hamming space

Let q = {0, 1, . . . , q − 1} be an alphabet with q ≥ 3 symbols w ∈ qn is a words of length n. d(u, v) := |{s | us = vs}| is the Hamming distance between u, v ∈ qn This makes qn into a metric space, the Hamming space H(n, q).

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 4

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Codes

A subset C ⊆ qn is a code. with minimum distance min{d(u, v) | u, v ∈ C, u = v}. Aq(n, d) := maximum cardinality of C ⊆ qn with min. distance d. Goal: find upper bounds for Aq(n, d). Note: Aq(n, d) = α(Gq(n, d)) with vertices qn and uv edge when d(u, v) < d. Bad news: The graph Gq(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

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SLIDE 5

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Codes

A subset C ⊆ qn is a code. with minimum distance min{d(u, v) | u, v ∈ C, u = v}. Aq(n, d) := maximum cardinality of C ⊆ qn with min. distance d. Goal: find upper bounds for Aq(n, d). Note: Aq(n, d) = α(Gq(n, d)) with vertices qn and uv edge when d(u, v) < d. Bad news: The graph Gq(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

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SLIDE 6

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Codes

A subset C ⊆ qn is a code. with minimum distance min{d(u, v) | u, v ∈ C, u = v}. Aq(n, d) := maximum cardinality of C ⊆ qn with min. distance d. Goal: find upper bounds for Aq(n, d). Note: Aq(n, d) = α(Gq(n, d)) with vertices qn and uv edge when d(u, v) < d. Bad news: The graph Gq(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

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SLIDE 7

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Hamming Scheme

We need the qn × qn 0–1 matrices A0, . . . , An given by: (Ak)u,v :=

  • 1

if d(u, v) = k,

  • therwise,

for u, v ∈ qn. 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

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SLIDE 8

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Inner distribution

The inner distribution (x0, x1, . . . , xn) of a code C is given by xi := (χC)TAiχC |C| . xi ≥ 0, x0 = 1, x1 = x2 = . . . = xd−1 = 0, |C| = x0 + x1 + . . . + xn, Delsarte constraints...

  • i xiKj(i) ≥ 0 for j = 0, . . . , n,

Kj(i) = j

r=0(−1)r(q − 1)j−ri r

n−i

j−r

  • are Krawtchouk

polynomials.

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 9

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Inner distribution

The inner distribution (x0, x1, . . . , xn) of a code C is given by xi := (χC)TAiχC |C| . xi ≥ 0, x0 = 1, x1 = x2 = . . . = xd−1 = 0, |C| = x0 + x1 + . . . + xn, Delsarte constraints...

  • i xiKj(i) ≥ 0 for j = 0, . . . , n,

Kj(i) = j

r=0(−1)r(q − 1)j−ri r

n−i

j−r

  • are Krawtchouk

polynomials.

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 10

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Inner distribution

The inner distribution (x0, x1, . . . , xn) of a code C is given by xi := (χC)TAiχC |C| . xi ≥ 0, x0 = 1, x1 = x2 = . . . = xd−1 = 0, |C| = x0 + x1 + . . . + xn, Delsarte constraints...

  • i xiKj(i) ≥ 0 for j = 0, . . . , n,

Kj(i) = j

r=0(−1)r(q − 1)j−ri r

n−i

j−r

  • are Krawtchouk

polynomials.

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 11

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Inner distribution

The inner distribution (x0, x1, . . . , xn) of a code C is given by xi := (χC)TAiχC |C| . xi ≥ 0, x0 = 1, x1 = x2 = . . . = xd−1 = 0, |C| = x0 + x1 + . . . + xn, Delsarte constraints...

  • i xiKj(i) ≥ 0 for j = 0, . . . , n,

Kj(i) = j

r=0(−1)r(q − 1)j−ri r

n−i

j−r

  • are Krawtchouk

polynomials.

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 12

Introduction Method of Schrijver Computational results Nonbinary codes Method of Delsarte

Delsarte relations

The Delsarte inequalities

  • i

xiKj(i) ≥ 0, for j = 0, . . . , n are equivalent to

  • i

xiAi · (vi)−1 is positive semidefinite, where vi is the number of nonzero entries of Ai.

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results A semidefinite program

Triples of words

Consider ordered triples of words (u, v, w) ∈ qn × qn × qn. i := d(u, v) j := d(u, w) t := #s with us = vs and us = ws p := #s with us = vs = ws. 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

4

  • tuples (i, j, t, p).
  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results A semidefinite program

Terwilliger algebra

For each tuple (i, j, t, p) define 0–1 matrix Mt,p

i,j :

(Mt,p

i,j )u,v :=

  • 1

if d(u, v, w) = (i, j, t, p),

  • therwise.

The linear span A := {

  • i,j,t,p

xt,p

i,j Mt,p i,j | xt,p i,j ∈ C}

is the Terwilliger algebra. Facts: A is those matrices invariant under permutating rows and columns by symmetries fixing 0. dim(A) = n+4

4

  • .
  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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SLIDE 15

Introduction Method of Schrijver Computational results A semidefinite program

Counting triples

In code C, we count triples of each type. xt,p

i,j := 1

|C|#triples (u, v, w) ∈ C3 with d(u, v, w) = (i, j, t, p). Observe: xt,p

i,j ≥ 0, x0,0 0,0 = 1

xt,p

i,j = 0 when {i, j, i + j − t − p} ∩ 1, 2, . . . , d − 1 = ∅

xt,p

i,j = xt′,p′ i′,j′ when t − p = t′ − p′ and

{i, j, i + j − t − p} = {i′, j′, i′ + j′ − t′ − p′}. x0,0

0,0 + x0,0 1,0 + · · · + x0,0 n,0 = |C|.

  • ther. . .
  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results A semidefinite program

Analogue of Delsarte relations

Analogous to Delsarte inequalities, the matrices R′ :=

  • i,j,t,p

xt,p

i,j (γt,p i,j )−1Mt,p i,j ,

R′′ :=

  • i,j,t,p

(x0,0

i+j−y−p,0(γ0,0 i+j−t−p,0)−1 − xt,p i,j (γt,p i,j )−1)Mt,p i,j ,

are positive semidefinite.

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results A semidefinite program

Proof

R′ =

  • σ|0∈σC

(χσC)TχσC · #autom |C| · qn R′′ =

  • σ|0∈σC

(χσC)TχσC · #autom |C| · qn

  • Mt,p

i,j , (χσC)TχσC

= #(0, u, v) ∈ (σC)3 with d(0, u, v) = (i, j, t, p)

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results A semidefinite program

Proof

R′ =

  • σ|0∈σC

(χσC)TχσC · #autom |C| · qn R′′ =

  • σ|0∈σC

(χσC)TχσC · #autom |C| · qn

  • Mt,p

i,j , (χσC)TχσC

= #(0, u, v) ∈ (σC)3 with d(0, u, v) = (i, j, t, p)

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results A semidefinite program

SDP

We obtain the following semidefinite programming bound for Aq(n, d): max

  • k

x0,0

k,0

s.t. xt,p

i,j ≥ 0, x0,0 0,0 = 1,

xt,p

i,j = 0 if i ∈ {1, 2, . . . , d − 1}

xt,p

i,j = xt′,p′ i′,j′ if t − p = t′ − p′ and

{i, j, i + j − t − p} = {i′, j′, i′ + j′ − t′ − p′} R, R′ are PSD.

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results A semidefinite program

Block diagonalization

For suitable unitary matrix U, the algebra U∗AU is all matrices    B1 · · · . . . ... · · · Bm    where each Bs is a block diagonal matrix    Cs · · · . . . ... · · · Cs    with repeated identical square matrices Cs on the diagonal. Block are indexed by pairs (a, k) with 0 ≤ a ≤ k ≤ n +a −k Block (a, k) has n

a

  • (q − 2)a[

n−a

k−a

  • −

n−a

n−a−1

  • ] copies
  • f size n + a + 1 − 2k.
  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results Tables

q = 3

best best upper lower new bound bound upper previously Delsarte n d known bound known bound 12 4 4374 6839 7029 7029 13 4 8019 19270 19682 19683 14 4 24057 54774 59046 59049 12 5 729 1557 1562 1562 13 5 2187 4078 4163 4163 14 5 6561 10624 10736 10736 13 6 729 1449 1562 1562 14 6 2187 3660 3885 4163 14 7 243 805 836 836 13 8 42 95 103 103 14 9 31 62 66 81

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results Tables

q = 4

best best upper lower new bound bound upper previously Delsarte n d known bound known bound 7 4 128 169 179 179 8 4 320 611 614 614 9 4 1024 2314 2340 2340 10 4 4096 8951 9360 9362 10 5 1024 2045 2048 2145 10 6 256 496 512 512 11 6 1024 1780 2048 2048 12 6 4096 5864 6241 6241 12 7 256 1167 1280 1280

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes

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Introduction Method of Schrijver Computational results Tables

q = 5

best best upper lower new bound bound upper previously Delsarte n d known bound known bound 7 4 250 545 554 625 7 5 53 108 125 125 8 5 160 485 554 625 9 5 625 2152 2291 2291 10 5 3125 9559 9672 9672 11 5 15625 44379 44642 44642 10 6 625 1855 1875 1875 11 6 3125 8840 9375 9375

  • D. Gijswijt, A. Schrijver, H. Tanaka

SDP bounds for codes