### Abstract

Let A(n,d) be the maximum number of 0, 1 words of length n , any two having Hamming distance at least d. It is proved that A(20,8)=256, which implies that the quadruply shortened Golay code is optimal. Moreover, it is shown that A(18,6) ≤ 673, A(19,6) ≤ 1237, A(20,6) ≤ 2279, A(23,6) ≤ 13674, A(19,8) ≤ 135, A(25,8) 5421, A(26,8) ≤ 9275, A(27,8) ≤ 17099, A(21,10) ≤ 47, A(22,10) ≤ 84, A(24,10) ≤ 268, A(25,10) ≤ 466, A(26,10) ≤ 836, A(27,10) ≤ 1585, A(28,10) ≤ 2817, A(25,12) ≤ 55, and A(26,12) ≤96. The method is based on the positive semidefiniteness of matrices derived from quadruples of words. This can be put as constraint in a semidefinite program, whose optimum value is an upper bound for A(n,d). The order of the matrices involved is huge. However, the semidefinite program is highly symmetric, by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry. By block diagonalizing this algebra, the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of n and d.

Original language | English (US) |
---|---|

Article number | 6142090 |

Pages (from-to) | 2697-2705 |

Number of pages | 9 |

Journal | IEEE Transactions on Information Theory |

Volume | 58 |

Issue number | 5 |

DOIs | |

State | Published - May 2012 |

### Fingerprint

### Keywords

- Algebra
- code
- error-correcting
- programming
- semidefinite

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Cite this

*IEEE Transactions on Information Theory*,

*58*(5), 2697-2705. [6142090]. https://doi.org/10.1109/TIT.2012.2184845

**Semidefinite code bounds based on quadruple distances.** / Gijswijt, Dion C.; Mittelmann, Hans; Schrijver, Alexander.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 58, no. 5, 6142090, pp. 2697-2705. https://doi.org/10.1109/TIT.2012.2184845

}

TY - JOUR

T1 - Semidefinite code bounds based on quadruple distances

AU - Gijswijt, Dion C.

AU - Mittelmann, Hans

AU - Schrijver, Alexander

PY - 2012/5

Y1 - 2012/5

N2 - Let A(n,d) be the maximum number of 0, 1 words of length n , any two having Hamming distance at least d. It is proved that A(20,8)=256, which implies that the quadruply shortened Golay code is optimal. Moreover, it is shown that A(18,6) ≤ 673, A(19,6) ≤ 1237, A(20,6) ≤ 2279, A(23,6) ≤ 13674, A(19,8) ≤ 135, A(25,8) 5421, A(26,8) ≤ 9275, A(27,8) ≤ 17099, A(21,10) ≤ 47, A(22,10) ≤ 84, A(24,10) ≤ 268, A(25,10) ≤ 466, A(26,10) ≤ 836, A(27,10) ≤ 1585, A(28,10) ≤ 2817, A(25,12) ≤ 55, and A(26,12) ≤96. The method is based on the positive semidefiniteness of matrices derived from quadruples of words. This can be put as constraint in a semidefinite program, whose optimum value is an upper bound for A(n,d). The order of the matrices involved is huge. However, the semidefinite program is highly symmetric, by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry. By block diagonalizing this algebra, the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of n and d.

AB - Let A(n,d) be the maximum number of 0, 1 words of length n , any two having Hamming distance at least d. It is proved that A(20,8)=256, which implies that the quadruply shortened Golay code is optimal. Moreover, it is shown that A(18,6) ≤ 673, A(19,6) ≤ 1237, A(20,6) ≤ 2279, A(23,6) ≤ 13674, A(19,8) ≤ 135, A(25,8) 5421, A(26,8) ≤ 9275, A(27,8) ≤ 17099, A(21,10) ≤ 47, A(22,10) ≤ 84, A(24,10) ≤ 268, A(25,10) ≤ 466, A(26,10) ≤ 836, A(27,10) ≤ 1585, A(28,10) ≤ 2817, A(25,12) ≤ 55, and A(26,12) ≤96. The method is based on the positive semidefiniteness of matrices derived from quadruples of words. This can be put as constraint in a semidefinite program, whose optimum value is an upper bound for A(n,d). The order of the matrices involved is huge. However, the semidefinite program is highly symmetric, by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry. By block diagonalizing this algebra, the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of n and d.

KW - Algebra

KW - code

KW - error-correcting

KW - programming

KW - semidefinite

UR - http://www.scopus.com/inward/record.url?scp=84860245290&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860245290&partnerID=8YFLogxK

U2 - 10.1109/TIT.2012.2184845

DO - 10.1109/TIT.2012.2184845

M3 - Article

AN - SCOPUS:84860245290

VL - 58

SP - 2697

EP - 2705

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 5

M1 - 6142090

ER -