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.
We exhibit two explicit circulant graphs of prime order $251$ that are $K_4$-free and have independence number $19$. Consequently \[R(4,20)\ge 252.\] These improve the bound $R(4,20)\ge 237$ given by Nagda, Raghavan, Thakurta and the long standing bound $R(4,21)\ge 242$ recorded in Radziszowski's dynamic survey. The gr...
We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - \epsilon$ is $(p, C_{p,q}/\epsilon)$-average-radius list-decodable with probability at least $1 - q^{-\Omega(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/\epsilon$ codewor...
V. Guruswami, Shi-Lun Li, Mihir Singhal· 2 citations· ⚡1
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 $s_2(N,\rho)$ be the smallest size of a $\rho$-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 n...
A. Davydov, S. Marcugini, F. Pambianco et al.· 0 citations
Let $C \le \mathbb{F}_q^n$ be a uniformly random $\mathbb{F}_q$-linear code of rate $1 - h_q(\rho) - \varepsilon$, and let $L^*(C,\rho)$ be the least $L$ such that every Hamming ball of relative radius $\rho$ contains at most $L$ codewords of $C$. That $L^* = \Theta_{q,\rho}(1/\varepsilon)$ has been known since work of...
Let $\mathrm{RM}_q(r,m)$ denote the generalized Reed--Muller code of order $r$ and length $q^m$ over the finite field $\mathbb{F}_q$. We completely determine the weight spectra of $\mathrm{RM}_3(2m-i,m)$ for $i=3,4$, $\mathrm{RM}_4(3m-i,m)$ for $i=2,3$, $\mathrm{RM}_5(4m-4,m)$, and $\mathrm{RM}_7(6m-4,m)$, in the range...
Min-Jia Shi, Zhao-Kang Xing, Patrick Solé· 0 citations
For an $[n,K]$ Reed--Solomon code, the covering radius is $n-K$. Gandikota, Ghazi, and Grigorescu proved deterministic NP-hardness of bounded-distance decoding when the decoding radius is $d$ below the covering radius for every $1\le d\le c\log n/\log\log n$, where $c>0$ is an absolute constant. We prove that, for ever...
Da-Qing Wan, Jun Zhang· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
