Toroidal boards and code covering

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.