Semidefinite code bounds based on quadruple distances

Dion C. Gijswijt, Hans Mittelmann, Alexander Schrijver

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


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
Issue number5
StatePublished - May 2012


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

ASJC Scopus subject areas

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


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