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Toroidal boards and code covering. (English) Zbl 07198013
Summary: 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 \(\mathbb{F}^3_q\) is the set of elements of \(\mathbb{F}^3_q\) with Hamming distance to \(V\) at most \(r\). We define \(c(q)\) to be the size of a minimum covering of \(\mathbb{F}^3_q\) by radius-1 extended balls. We define a semiqueen to be 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 \(q\geq 7\), \(c(q)=\xi_D(q-1)+2\). Moreover, our proof exhibits a method to build such covers of \(\mathbb{F}^3_q\) 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.
94B Theory of error-correcting codes and error-detecting codes
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