Semidefinite bounds for mixed binary/ternary codes

Abstract

For nonnegative integers n2,n3 and d, let N(n2,n3,d) denote the maximum cardinality of a code of length n2+n3, with n2 binary coordinates and n3 ternary coordinates (in this order) and with minimum distance at least d. For a nonnegative integer k, let Ck denote the collection of codes of cardinality at most k. For D∈Ck, define S(D)≔{C∈Ck∣D⊆C,|D|+2|C∖D|≤k}. Then N(n2,n3,d) is upper bounded by the maximum value of ∑v∈[2]n2[3]n3x({v}), where x is a function Ck→R such that x(∅)=1 and x(C)=0 if C has minimum distance less than d, and such that the S(D)×S(D) matrix (x(C∪C′))C,C′∈S(D) is positive semidefinite for each D∈Ck. By exploiting symmetry, the semidefinite programming problem for the case k=3 is reduced using representation theory. It yields 135 new upper bounds that are provided in tables.