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- W3134762792 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 $ntimes n$ toroidal board necessary to cover the entire board except possibly for the southwest-northeast diagonal. We prove that, for $qge 7$, $c(q)=xi_D(q-1)+2$. 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." @default.
- W3134762792 created "2021-03-15" @default.
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- W3134762792 date "2015-10-28" @default.
- W3134762792 modified "2023-09-24" @default.
- W3134762792 title "Toroidal boards and code covering" @default.
- W3134762792 hasPublicationYear "2015" @default.
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