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Covering 1024 syndromes with 50 columns
Abstract
We exhibit a binary linear [50,40]2 code of covering radius $2$, so ℓ2(10,2)≤ 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 QM22 propagates it to exhaustively verified codes of lengths $815$ and $1631$ at r=18 and r=20, and to the family n=51· 2r/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.
Related
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BibTeX
@misc{wu2026covering,
title = {{Covering 1024 syndromes with 50 columns}},
author = {Stephen Wu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27494},
eprint = {2608.27494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27494},
}
RIS
TY - UNPB
TI - Covering 1024 syndromes with 50 columns
AU - Stephen Wu
PY - 2026
DA - 2026/08/26
PB - arXiv
DO - 10.48550/arxiv.2608.27494
UR - https://arxiv.org/abs/2608.27494
AB - We exhibit a binary linear [50,40]_2 code of covering radius $2$, so ℓ_2(10,2)≤ 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 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· 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.
ER -