Abstract
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 mathcal{C}_k be the collection of codes of cardinality at most k. Then A_q(n,d) is at most the maximum value of sum _{vin [q]^n}x({v}), where x is a function mathcal{C}_4rightarrow {mathbb {R}}_+ such that x(emptyset )=1 and x(C)=!0 if C has minimum distance less than d, and such that the mathcal{C}_2times mathcal{C}_2 matrix (x(Ccup C'))_{C,C'in mathcal{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)le 176, A_4(7,3)le 596, A_4(7,4)le 155, A_5(7,4)le 489, and A_5(7,5)le 87.
Highlights
Let Z+ denote the set of nonnegative integers, and denote [m] = {1, . . . , m}, for any m ∈ Z+
We will assume throughout that q ≥ 2. (If you prefer {0, 1, . . . , q − 1} as alphabet, take the letters mod q.) While this paper is mainly meant to handle the case q ≥ 3, the results hold for q = 2
The minimum distance of a code C is the minimum of dH (v, w) taken over distinct
Summary
The optimization problem (2) is huge, but, with methods from representation theory, can be reduced to a size bounded by a polynomial in n, with entries (i.e., coefficients) being polynomials in q This makes it possible to solve (2) by semidefinite programming for some moderate values of n, d, and q, leading to improvements of best known upper bounds for Aq (n, d). Let H be the wreath product Sqn Sn. For each k, the group H acts naturally on Ck, maintaining minimum distances and cardinalities of elements of Ck (being codes). The optimum x can be replaced by the average of all g · x (over all g ∈ H ), by the convexity of the set of positive semidefinite matrices This makes the optimum solution H -invariant.
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