Semidefinite bounds for nonbinary codes based on quadruples - Litjens, Bart; Polak, Sven; Schrijver, Alexander (Designs, codes, and cryptography, 2017)
For nonnegative integers q , n , d , let A q ( n , d ) denote the maximum cardinality of a code of length n over an alphabet [ q ] with q letters and with minimum distance at least d . We consider the following upper bound on A q ( n , d ) . For any k , let C k be the collection of codes of cardinality at most k . Then A q ( n , d ) is at most the maximum value of ∑ v ∈ [ q ] n x ( { v } ) , where x is a function C 4 → R + such that x ( ∅ ) = 1 and x ( C ) = 0 if C has minimum distance less than d , and such that the C 2 × C 2 matrix ( x ( C ∪ C ′ ) ) C , C ′ ∈ C 2 is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n . It yields the new upper bounds A 4 ( 6 , 3 ) ≤ 176 , A 4 ( 7 , 3 ) ≤ 596 , A 4 ( 7 , 4 ) ≤ 155 , A 5 ( 7 , 4 ) ≤ 489 , and A 5 ( 7 , 5 ) ≤ 87 .