PreprintCC BY 4.0arXiv
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≥ 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 Ad+1=⋯=Ad+t=0 for some t≥ 1 and k≥ n−2d+1, then Aw=0 for all w∈[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.
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BibTeX
@misc{he2026mirror,
title = {{A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes}},
author = {Xianmang He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20344},
eprint = {2609.20344},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20344},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes
AU - Xianmang He
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20344
UR - https://arxiv.org/abs/2609.20344
AB - 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≥ 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=⋯=A_d+t=0 for some t≥ 1 and k≥ n−2d+1, then A_w=0 for all w∈[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.
ER -