用50列覆盖1024种症候
Covering 1024 syndromes with 50 columns
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中文总结 AI 辅助
该研究构造了长度比Kaikkonen–Rosendahl长度少1的二元线性覆盖码，通过QM₂²构造法扩展出多种经穷举验证的码及渐近码族，相关资源可在指定链接获取。
中文摘要 AI 辅助
我们构造了一个覆盖半径为2的二元线性[50,40]_2码，因此ℓ₂(10,2)≤50，该长度比自2003年以来未被突破的Kaikkonen–Rosendahl长度51少1，且仍为Davydov–Marcugini–Pambianco的R=2族（arXiv:2511.02542）提供支撑。该新矩阵可划分为10个块，采用(2,0)-划分，因此QM₂²构造法将其扩展为在r=18和r=20时长度分别为815和1631的经穷举验证的码，以及渐近密度为2601/2048的n=51·2^(r/2−5)−1族。相关矩阵、验证工具和源代码位于https://github.com/pinproblems/covering/share/2026-08-24/（commit 736a38f）。
英文摘要
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.
