二元线性码重量分布的镜像消失带
A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes
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中文总结 AI 辅助
利用极小向量不相交支撑分解性质，在二元线性码重量分布中构造$2d$另一侧的镜像消失带，改进Chen-Xie界，将非零重量数上界降低$2t$。
中文摘要 AI 辅助
Chen和Xie最近利用Ashikhmin-Barg关于极小向量的引理证明了：每个满足$k=n-2d+2+v$（$v\ge 0$）的二元线性$[n,k,d]$码在区间$[2d-v, 2d-1]$内没有重量为$w$的码字。他们的论证使用了Ashikhmin和Barg（1998）建立的极小向量五个基本性质中的两个。在本注记中，我们利用第三个性质，即二元码中非极小码字的不相交支撑分解，在$2d$的另一侧生成一个“镜像”消失带：若对某个$t\ge 1$有$A_{d+1}=\cdots=A_{d+t}=0$且$k\ge n-2d+1$，则对所有$w\in[2d+1, 2d+t]$有$A_w=0$。结合这两个带，此类码的非零重量个数至多为$n-d-v-2t+1$，比Chen-Xie界$n-d+1-v$改进了$2t$。
英文摘要
Chen and Xie recently proved, using the Ashikhmin--Barg lemma on minimal vectors, that every binary linear $[n,k,d]$ code with $k=n-2d+2+v$ ($v\ge 0$) has no codewords of weight in the interval $[2d-v,\,2d-1]$. Their argument uses two of the five basic properties of minimal vectors established by Ashikhmin and Barg (1998). In this note we utilize the third property, the disjoint-support decomposition of non-minimal codewords in binary codes, to generate a \emph{mirror} vanishing band on the other side of $2d$: if $A_{d+1}=\cdots=A_{d+t}=0$ for some $t\ge 1$ and $k\ge n-2d+1$, then $A_w=0$ for all $w\in[2d+1,\,2d+t]$. Combining the two bands, the number of nonzero weights of such a code is at most $n-d-v-2t+1$, improving the Chen--Xie bound $n-d+1-v$ by $2t$.
