Covering 1024 syndromes with 50 columns

2026-08-26

AI Summary

本文通过构建一个二进制线性[50,40]_2码来覆盖1024个综合症，其覆盖半径为2，使用了新的矩阵分区方法以减少所需列数。

Abstract

We exhibit a binary linear $[50,40]_2$ code of covering radius $2$, so $\ell_2(10,2)\le 50$, one column below the Kaikkonen--Rosendahl length $51$ that has stood since 2003 and that still seeds the $R=2$ family of Davydov--Marcugini--Pambianco (arXiv:2511.02542). The new matrix admits a $(2,0)$-partition into ten blocks, so Construction $\mathrm{QM}_2^2$ propagates it to exhaustively verified codes of lengths $815$ and $1631$ at $r=18$ and $r=20$, and to the family $n=51\cdot 2^{r/2-5}-1$ of asymptotic density $2601/2048$. The matrices, verifiers, and source are at https://github.com/wustep/maths, pin problems/covering/share/2026-08-24/ at commit 736a38f.

Problem

• covering radius
• binary linear code
• syndromes

Innovation

• binary linear code
• covering radius
• partition
• asymptotic density
• code propagation