覆盖半径为2的二元码与射影空间中饱和集的进一步结果
Further results on binary codes of covering radius 2 and saturating sets in projective spaces
- Kharkevich Institute for Information Transmission Problems Russian Academy of Sciences（信息传输问题哈尔维奇研究所 俄罗斯科学院）
- Department of Mathematics and Computer Science, University of Perugia（佩鲁贾大学数学与计算机科学系）
- Independent（独立研究者）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造新的无限二元码族，改进覆盖半径为2的码长上界，并降低渐近覆盖密度至1.27002，从而改进Green问题40的常数上界。
AI中文摘要：
长度函数$\ell_2(r,R)$是余维数（冗余）为$r$、覆盖半径为$R$的二元线性码的最小长度。设$s_2(N,\rho)$为射影空间$\mathrm{PG}(N,2)$中$\rho$-饱和集的最小大小。已知$\ell_2(r,R)=s_2(r-1,R-1)$。我们获得了关于$\ell_2(r,2)$的以下新的上界，与先前已知的最佳上界相比，这些上界带来了减少量$\Delta(r,2)$：对于$r=2t$，$r=10,18,20$以及$r\ge28$，有$\ell_2(r,2)=s_2(r-1,1)\le51\cdot2^{r/2-5}-1$；$\Delta(r,2)=2^{r/2-5}$。为了获得这些界，我们构造了一个新的无限码族，使用了覆盖码的$q^m$-拼接构造的不同版本；其中一些版本是在本文中提出的。我们还获得了某些码的奇偶校验矩阵列集合的新的有用划分。由新码族提供的渐近覆盖密度$\overline{\mu}(2)\le1.27002$小于先前已知的值，并给出了Green开放问题40中常数$f(2)$的新上界$f(2)\le1.27002$。
英文摘要：
The length function $\ell_2(r,R)$ is the smallest length of a binary linear code with codimension (redundancy) $r$ and covering radius $R$. Let $s_2(N,ρ)$ be the smallest size of a $ρ$-saturating set in the projective space $\mathrm{PG}(N,2)$. It is known that $\ell_2(r,R)=s_2(r-1,R-1)$. We obtain the following new upper bounds on $\ell_2(r,2)$, which yield a decrease $Δ(r,2)$ compared to the best previously known upper bounds: $r=2t,r=10,18,20$ and $r\ge28,\ell_2(r,2)=s_2(r-1,1)\le51\cdot2^{r/2-5}-1;Δ(r,2)=2^{r/2-5}$. To obtain these bounds, we construct a new infinite code family, using distinct versions of the $q^m$-concatenating constructions of covering codes; some of these versions are proposed in this paper. We also obtain new useful partitions of column sets of parity check matrices of some codes. The asymptotic covering density $\overlineμ(2)\le1.27002$, provided by the codes of the new family, is smaller than previously known one and gives rise to the new upper bound $f(2)\le1.27002$ on the constant $f(2)$ of the Green's Open Problem 40.
