AI 中文总结
本文研究广义打包-覆盖猜想，证明任意有限域上冗余度至多十四的线性码均满足该猜想，并利用辅助码准则证明广义汉明重量界，同时分析二元本原BCH码的打包半径与覆盖半径关系。
AI 中文摘要
广义打包-覆盖猜想询问：在每个阶数下，线性码的打包半径是否至多等于其覆盖半径。我们证明了在任意有限域上，冗余度至多为十四的每个线性码都满足该猜想，将先前已建立的冗余度七的范围进行了扩展。我们还证明了当字母表大小 q 满足 q≥R_t(C) 时，广义汉明重量界 d_t(C)≤2R_t(C)+1 成立，这使用了辅助码准则，该准则将伴随式空间的覆盖性质转化为重量界。对于二元本原BCH码，一旦扩展度足够大，对于每个固定的错误参数和阶数（两者均至少为2），打包半径严格小于覆盖半径；这由现有的覆盖界得出。
英文摘要
The generalized packing--covering conjecture asks whether, at every order, the packing radius of a linear code is at most its covering radius. We prove the conjecture for every linear code of redundancy at most fourteen over every finite field, extending the previously established range of redundancies at most seven. We also prove the bound on generalized Hamming weights $d_t(C)\le2R_t(C)+1$ whenever the alphabet size $q$ satisfies $q\ge R_t(C)$. Both results use an auxiliary-code criterion that converts a covering property in the syndrome space into a weight bound. For binary primitive BCH codes, the packing radius is strictly smaller than the covering radius for every fixed error parameter and order, both at least two, once the extension degree is sufficiently large; this follows from existing covering bounds.
Comments10 pages, 2 tables. Revised exposition and references; clarified the auxiliary-code method and improved presentation
