Semidefinite Programming Bounds for Constant-Weight Codes

Abstract

For nonnegative integers n, d, w, let A(n, d, w) be the maximum size of a code C ⊆ F n 2 with constant weight w and minimum distance at least d. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on A(n, d, w). The new upper bounds imply that A(22, 8, 10) = 616 and A(22, 8, 11) = 672. Lower bounds on A(22, 8, 10) and A (22,8,11) are obtained from the (n, d) = (22, 7) shortened Golay code of size 2048. It can be concluded that the shortened Golay code is a union of constant weight w codes of sizes A(22, 8, w).

Uniqueness of codes using semidefinite programming

“…In [15], it is found that A 3 (23, 8, 11) = 1288, matching the known lower bound and thereby proving that A(23, 8, 11) = 1288. Similarly, in [13], the upper bounds A(22, 8, 10) ≤ 616 and A(22, 8, 11) ≤ 672 are obtained, which imply A(22, 8, 10) = 616 and A(22, 8, 11) = 672. The latter two upper bounds are in fact instances of the bound B 4 (n, d, w), which is a bound in between A 3 (n, d, w) and A 4 (n, d, w).…”

“…It can be seen (cf. [6,13]) that the nonnegativity condition on x, and hence on y, is already imposed by positive semidefiniteness of all matrices M k,D (x). When solving the semidefinite program with a computer, we add the constraints y ω ≥ 0 seperately, by adding 1 × 1 blocks (y ω ) which are required to be positive semidefinite.…”

“…A binary code C ⊆ F n 2 and d min (C) ≥ d is called an (n, d)-code. Using semidefinite programming, some upper bounds on A(n, d, w) have recently been obtained that are equal to the best known lower bounds: it has been established that A(23, 8, 11) = 1288 (see [15]), and that A(22, 8, 11) = 672 and A(22, 8, 10) = 616 (see [13]). We show using the output of the corresponding semidefinite programs that the codes of maximum size are unique (up to coordinate permutations) for these n, d, w.…”

Semidefinite programming bounds for Lee codes

“…Proofs are omitted, but for more information, the reader can consult Sagan [21]. The content of this section is the same as Section 2 of [15,19], so readers who are familiar with one of these papers can safely skip this section.…”

“…For binary codes equipped with the Hamming distance, the Delsarte bound was generalized to a semidefinite programming bound based on triples of codewords by A. Schrijver [23], and later to a quadruple bound by Gijswijt, Mittelmann and Schrijver [10]. Also, nonbinary codes with the Hamming distance have been considered [11,15], codes with mixed alphabets [16] and constant weight (binary) codes [19,23]. In [5], the authors mention the possibility of applying semidefinite programming to Lee codes and they state that to their best knowledge, such bounds for Lee codes using triples have not yet been studied.…”