Binary code weights have new symmetrical gaps beyond known limits
A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes
Information Theory
Summary
Binary linear codes are used in digital communication to detect and correct errors. The authors build on previous work that showed certain weights (numbers of ones) of codewords cannot appear in specific ranges. They discover a new, matching gap in the weight spectrum on the opposite side of a key weight, which further restricts where codewords can appear. Together, these new and old results tighten the rules on possible codeword weights, helping to better understand the structure of these codes.
What this means in practice
- •For coding theorists: Refine bounds on weight distributions to improve classification and design of binary codes with specific parameters.
- •For error correction engineers: Use tighter constraints on codeword weights to optimize error correction strategies for communication systems.
A theory result. No direct application yet.
Authors
Xianmang He
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 $2t$.