For nonnegative integers q, n, d, let $$A_q(n,d)$$Aq(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 $$A_q(n,d)$$Aq(n,d). For any k, let $$\mathcal{C}_k$$Ck be the collection of codes of cardinality at most k. Then $$A_q(n,d)$$Aq(n,d) is at most the maximum value of $$\sum _{v\in [q]^n}x(\{v\})$$∑v∈[q]nx({v}), where x is a function $$\mathcal{C}_4\rightarrow {\mathbb {R}}_+$$C4→R+ such that $$x(\emptyset )=1$$x(∅)=1 and $$x(C)=\!0$$x(C)=0 if C has minimum distance less than d, and such that the $$\mathcal{C}_2\times \mathcal{C}_2$$C2×C2 matrix $$(x(C\cup C'))_{C,C'\in \mathcal{C}_2}$$(x(C∪C′))C,C′∈C2 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 $$A_4(6,3)\le 176$$A4(6,3)≤176, $$A_4(7,3)\le 596$$A4(7,3)≤596, $$A_4(7,4)\le 155$$A4(7,4)≤155, $$A_5(7,4)\le 489$$A5(7,4)≤489, and $$A_5(7,5)\le 87$$A5(7,5)≤87.

中文翻译：

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基于四元组的非二进制码的半定界

对于非负整数 q, n, d, 让 $$A_q(n,d)$$Aq(n,d) 表示长度为 n 的代码在具有 q 个字母和最小距离的字母表 [q] 上的最大基数最少 D. 我们考虑以下 $$A_q(n,d)$$Aq(n,d) 的上限。对于任何 k，令 $$\mathcal{C}_k$$Ck 是至多 k 的基数代码的集合。那么$$A_q(n,d)$$Aq(n,d)至多是$$\sum _{v\in [q]^n}x(\{v\})$$∑的最大值v∈[q]nx({v})，其中 x 是一个函数 $$\mathcal{C}_4\rightarrow {\mathbb {R}}_+$$C4→R+ 使得 $$x(\emptyset ) =1$$x(∅)=1 并且 $$x(C)=\!0$$x(C)=0 如果 C 的最小距离小于 d，并且 $$\mathcal{C}_2 \times \mathcal{C}_2$$C2×C2 矩阵 $$(x(C\cup C'))_{C,C'\in \mathcal{C}_2}$$(x(C∪C' ))C,C′∈C2 是半正定的。由问题的对称性，我们可以应用表示理论将问题简化为阶数以 n 中的多项式为界的半定规划问题。它产生新的上限 $$A_4(6,3)\le 176$$A4(6,3)≤176, $$A_4(7,3)\le 596$$A4(7,3)≤596, $$A_4(7,4)\le 155$$A4(7,4)≤155，$$A_5(7,4)\le 489$$A5(7,4)≤489，以及 $$A_5(7) ,5)\le 87$$A5(7,5)≤87。