17 Citations (Scopus)

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 languageEnglish (US)
Article number6142090
Pages (from-to)2697-2705
Number of pages9
JournalIEEE Transactions on Information Theory
Volume58
Issue number5
DOIs
StatePublished - May 2012

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Algebra
Hamming distance
Values
programming
software

Keywords

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

ASJC Scopus subject areas

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

Cite this

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

In: IEEE Transactions on Information Theory, Vol. 58, No. 5, 6142090, 05.2012, p. 2697-2705.

Research output: Contribution to journalArticle

Gijswijt, Dion C. ; Mittelmann, Hans ; Schrijver, Alexander. / Semidefinite code bounds based on quadruple distances. In: IEEE Transactions on Information Theory. 2012 ; Vol. 58, No. 5. pp. 2697-2705.
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