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 new upper bounds on $\ell_2(r,2)$, which yield a decrease $\Delta(r,2)$ compared to the best previously known upper bounds: $r=2t,r=10,18,20$ and $r\ge28,\ell_2(r,2)=s_2(r-1,1)\le51\cdot2^{r/2-5}-1;\Delta(r,2)=2^{r/2-5}$. 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 $\overline{\mu}(2)\le1.27002$, 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.
Let $\text{RM}(r,n)$ be the $r$th order Reed-Mullet code of length $2^n$, and let $\rho(r,n)$ denote its covering radius. It is known that $242\le\rho(1,9)\le 244$, and the determination of $\rho(1,9)$ is a long-standing open question. We give a computer-assisted proof that $\rho(1,9)=242$, leveraging the classificatio...
The set-coloring Ramsey number $R(k;r,s)$ is the least $N$ such that every assignment of an $s$-element subset of $[r]$ to each edge of $K_N$ yields a copy of $K_k$ whose edges share a common color. For every fixed prime power $q$, we construct infinitely many positive integer triples $(r,j,s)$ with $j\sim(q-1)^{-2/3}r...
Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the associated congruent-nu...
Let $r_1,\ldots,r_t$ be positive integers, and let $\mathcal A$ be the set of integers of the form $2^{k_1^{r_1}}+\cdots+2^{k_t^{r_t}}$, where $k_1,\ldots,k_t$ are positive integers. For any positive integer $n$, let $r(n)$ be the number of representations $n=p+a$ with $p$ an odd prime and $a\in \mathcal A$. We prove t...
Let $A_q(n,d)$ denote the largest size of a code of length $n$ and minimum Hamming distance $d$ over an alphabet of $q$ symbols. For fixed $q\ge2$ and $d\ge4$ we construct codes with redundancy $n-\log_q A_q(n,d)\le(d-3)\log_q n+O(\sqrt{\log n})$ for all large $n$. No algebraic structure is imposed on the alphabet, and...
Let $q\geq3$ be odd and put $m_r=\lfloor r\log_2q\rfloor$. We count parity sequences with exactly $r$ entries equal to $1$ for which the coefficient in the $qx+1$ iteration first drops below $1$ at step $m_r+1$. Their number $a_q(r)$ also counts the corresponding residue classes modulo $2^{m_r+1}$. By classifying incre...
Mike Winkler, Ph.D.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
