A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes

Abstract

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 $2t$.