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 .
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