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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.
MSC:
94B Theory of error-correcting codes and error-detecting codes
Software:
CPLEX; GLPK; Gurobi
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References:
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