列表可译码线性码的Griesmer型界
A Griesmer-Type Bound for List-Decodable Linear Codes
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中文总结 AI 辅助
研究针对列表可译码线性码，根据广义汉明重量推导列表译码半径上界，得出\(1\le L\le q - 1\)时相关线性码最小距离的结论，结合经典Griesmer界得块长下界，还构造特定码族说明Griesmer型界可改进Singleton型界。
中文摘要 AI 辅助
一个码\(C\subseteq F_q^n\)若半径为\(\tau\)的每个汉明球至多包含\(C\)的\(L\)个码字，则它是\((\tau,L)\)-列表可译码的。Singleton型界在\(L\)固定时约束半径和码率。本文根据广义汉明重量推导出列表译码半径的上界。结果表明，对于\(1\le L\le q - 1\)，每个\((\tau,L)\)-列表可译码的\(q\)元线性码的最小距离至少为\(\tau+\left\lfloor \frac{\tau}{L}\right\rfloor+1\)。结合此下界与经典Griesmer界得到块长的Griesmer型下界。对于\(q = 3^a\)，构造了明确的\(q\)元线性\([q + 3,2,q + 1]\)码族，它们是\((2q/3,2)\)-列表可译码且达到Griesmer型界等式，未达到Singleton型界，说明Griesmer型界可严格改进Singleton型界。
英文摘要
A code $C\subseteq F_q^n$ is $(τ,L)$-list-decodable if every Hamming ball of radius $τ$ contains at most $L$ codewords of $C$. Here $τ$ is the list-decoding radius, and $L$ is the list size. Singleton-type bounds constrain the radius and the rate when $L$ is fixed. These bounds are not the only possible constraints on list-decodable codes. In this paper, we derive an upper bound on the list-decoding radius in terms of generalized Hamming weights. As a consequence, for $1\le L\le q-1$, every $(τ,L)$-list-decodable $q$-ary linear code has minimum distance at least $ τ+\left\lfloor \fracτ{L}\right\rfloor+1. $ Combining this lower bound with the classical Griesmer bound gives a Griesmer-type lower bound on the block length. For $q=3^a$, we construct an explicit family of $q$-ary linear $[q+3,2,q+1]$ codes. These codes are $(2q/3,2)$-list-decodable and meet the Griesmer-type bound with equality. They do not attain the Singleton-type bound. Thus the Griesmer-type bound can be a strict improvement over the Singleton-type bound.
