字母大小为6和7的覆盖码K_q(n,R)的新上界
New upper and lower bounds on covering codes K_q(n,R) for alphabets of size 5 <= q <= 21
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中文总结 AI 辅助
本文针对q=6、7的覆盖码K_q(n,R)，通过聚焦局部搜索以构造解为种子，得到9个K_q(n,R)的新上界，是2011年以来q≥5时该类上界的首次改进，9个码均附带验证工具与结果。
中文摘要 AI 辅助
针对q∈{6,7}的覆盖码K_q(n,R)（即长度为n、覆盖半径为R的q元码的最小基数）标准界表中的9个条目，本文给出改进后的上界：K_6(7,3)≤232、K_6(8,3)≤1045、K_6(8,4)≤167、K_6(9,4)≤703、K_6(9,5)≤123、K_6(10,4)≤2951、K_6(10,5)≤610、K_7(8,4)≤329、K_7(9,4)≤1743。此前Keri表（2011年最后更新）中的最优界均来自通用构造（直和及相关乘积规则）而非显式搜索，据作者所知，这是2011年以来q≥5时K_q(n,R)上界的首次改进。新界通过以构造所得现有解为种子的聚焦局部搜索得到，9个码均在辅助文件中明确给出，附带独立验证工具，每个码均通过4种独立的穷举验证方法检查。
英文摘要
Let K_q(n,R) denote the minimum cardinality of a q-ary code of length n with covering radius R. We improve the known bounds on K_q(n,R) in 83 cases (82 distinct cells). On the upper-bound side we give 25 improved bounds for 5<=q<=15 -- twenty-four found by search and one propagated by monotonicity -- using two complementary methods: an engineered focused local search seeded with structural constructions, and a large-neighbourhood search driven by exact full-space coverage transforms that evaluates every candidate codeword position simultaneously. These are, to our knowledge, the first improvements to any upper bound on K_q(n,R) with q >= 5 since the 2011 revision of Keri's tables; several bounds decrease by more than 20%, e.g. K_6(8,4)<=166 (previously 216) and K_8(10,5)<=1883 (previously 2461). On the lower-bound side we give 58 improved bounds for 6<=q<=21, obtained from the semidefinite programming hierarchy of Gijswijt and Polak, whose published results cover q<=5, by combining an exact-arithmetic reimplementation of the reduced program with a multiprecision solution pipeline. Every new lower bound is certified by a rational dual solution validated by a standalone exact-arithmetic checker; no floating-point computation is part of the trusted base. The same pipeline also gives strong numerical evidence of limits: on a dozen further cells the certified value of the relaxation, which the solver reports as optimal to within its working precision, lies below the best known bound, indicating that no improvement is available there at this level of the hierarchy. One cell is improved from both sides: 441<=K_6(10,4)<=2751, previously 417--2952. All codes and certificates are provided in machine-readable form together with standalone verifiers.
发表机构
- Budapest University of Technology and Economics（布达佩斯技术与经济大学）
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