发表机构
National Yang Ming Chiao Tung University; The Hong Kong University of Science and Technology（国立阳明交通大学; 香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用Gauss相位将LCD码条件转化为线性约束，构建增强线性规划，在二元和三元码上系统强化了Hamming界，并取得四项严格改进。
AI 中文摘要
对于$q\in\set{2,3}$，我们证明：在有限域$\F_q$上，一个$k$维线性码是线性互补对偶（LCD）码，当且仅当其重量计数器的某个单位根值的模为$q^{k/2}$。我们将该值的相位，连同二元情形下的奇偶类型，转化为关于重量分布的精确线性约束，并将其纳入一个Gauss相位线性规划。该规划仅使用码及其对偶码的普通重量分布，并在通常的Hamming/MacWilliams约束基础上仅增加一组常数大小的分支方程，因此其规模与标准Hamming线性规划相近，同时保留了额外的算术信息。在经审计的二元和三元范围内的计算表明，该规划系统地强化了Hamming LCD松弛。在二元情形下，与已建立的混合联合重量计数器线性规划的比较产生了四项严格改进，每项都将基准上界降低1。每项严格比较均通过有理可行性证人和整数Farkas证书精确验证。
英文摘要
We derive linear programming (LP) bounds on the minimum distance of binary and ternary linear complementary dual (LCD) codes by imposing arithmetic constraints on their weight enumerators. Special values of the weight enumerator give finitely many Gauss phases, each of which yields linear equations in the ordinary weight-distribution variables. The resulting bounds strengthen the real-valued LCD counting LP without introducing additional variables; both the number of branches and the number of added equations per branch are bounded independently of the code length. Exact certificates establish strict improvements for 62 binary parameter pairs of length at most 20 and 39 ternary pairs of length at most 14. For four binary pairs, the bounds also improve the joint-weight-enumerator LP while using fewer variables per branch. These comparisons show that Gauss-phase information provides a compact strengthening of existing LP relaxations for LCD codes.
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