New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming

Abstract

We give a new upper bound on the maximum size A q (n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet of q 3 letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in n using semidefinite programming. For q = 3, 4, 5 this gives several improved upper bounds for concrete values of n and d. This work builds upon previous results of Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (2005) 2859-2866] on the Terwilliger algebra of the binary Hamming scheme.

A semidefinite programming hierarchy for packing problems in discrete geometry

“…Packing problem 2-point bound 3-point bound 4-point bound Binary codes Delsarte [11] Schrijver [35] Gijswijt, Mittelmann, Schrijver [16] q-ary codes Delsarte [11] Gijswijt, Schrijver, Tanaka [15] Constant weight codes Delsarte [11] Schrijver [35], Regts [32] Spherical codes…”

Numerical block diagonalization of matrix *-algebras with application to semidefinite programming

“…Some information on A S D P is given in Table 1. The instance corresponding to r = 7 was first solved in [12], by solving the partially reduced SDP problem (9), that in this case takes the form:…”

“…More recent applications are surveyed in [7,11,28] and include bounds on kissing numbers [2], bounds on crossing numbers in graphs [12,13], bounds on code sizes [9,17,24], truss topology design [3,10], quadratic assignment problems [14], etc.…”

Semidefinite bounds for nonbinary codes based on quadruples

“…With this, we found the following improvements on the known bounds for A q (n, d), with thanks to Hans D. Mittelmann for his help in solving the larger-sized problems. Best lower bound known New upper bound Best upper bound previously known 4 6 3 164 176 179 4 7 3 512 596 614 4 7 4 128 155 169 5 7 4 250 489 545 5 7 5 53 87 108 The best upper bounds previously known for A 4 (6, 3) and A 4 (7, 3) are Delsarte's linear programming bound [4]; the other three best upper bounds previously known were given by Gijswijt, Schrijver, and Tanaka [7]. We refer to the most invaluable tables maintained by Andries Brouwer [3] with the best known lower and upper bounds for the size of errorcorrecting codes (see also Bogdanova, Brouwer, Kapralov, and Östergård [1] and Bogdanova and Östergård [2] for studies of bounds for codes over alphabets of size q = 4 and q = 5, respectively).…”

“…Our present description gives a more conceptual and representation-theoretic approach to the method of [6]. A bound intermediate to the Delsarte bound and the currently investigated bound is based on considering functions x : C 3 → R + and the related matrices-see Schrijver [9] for binary codes and Gijswijt et al [7] for nonbinary codes.…”