Toroidal boards and code covering

TL;DR

This paper establishes a connection between covering problems in finite vector spaces and combinatorial coverings on toroidal chessboards, providing exact values and bounds for minimal coverings using new geometric and combinatorial methods.

Contribution

It introduces a novel method to construct minimal coverings of $qt$ from semiqueen coverings of toroidal boards, determining $c(q)$ for odd $q$ and improving bounds for even $q$.

Findings

Proves that for $q extgreater 6$, $c(q)=\xi_D(q-1)+2$.

Provides exact values of $c(q)$ for odd $q$.

Improves bounds for the even $q$ case.

Abstract

We denote by $\mathbb{F}_q$ the field with $q$ elements. A radius-$r$ extended ball with center in a $1$-dimensional vector subspace $V$ of $\mathbb{F}_q^3$ is the set of elements of $\mathbb{F}_q^3$ with Hamming distance to $V$ at most $r$. We define $c(q)$ as the size of a minimum covering of $\fqt$ by radius-$1$ extended balls. We define a semiqueen as a piece of a toroidal chessboard that extends the covering range of a rook by the southwest-northeast diagonal containing it. Let $\xi_D(n)$ be the minimum number of semiqueens of the $n\times n$ toroidal board necessary to cover the entire board except possibly for the southwest-northeast diagonal. We prove that, for $q\ge 7$, $c(q)=\xi_D(q-1)+2$. Moreover, our proof exhibits a method to build such covers of $\mathbb{F}_q^3$ from the semiqueen coverings of the board. With this new method, we determine $c(q)$ for the odd values of $q$ and improve both existing bounds for the even case.