Toroidal boards and code covering
Jo\~ao Paulo Costalonga

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 the field with elements. A radius- extended ball with center in a -dimensional vector subspace of is the set of elements of with Hamming distance to at most . We define as the size of a minimum covering of by radius- 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 be the minimum number of semiqueens of the toroidal board necessary to cover the entire board except possibly for the southwest-northeast diagonal. We prove that, for , . Moreover, our proof exhibits a method to build such covers of from the semiqueen coverings of the board. With this new method, we determine for the odd values of …
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Taxonomy
TopicsAdvanced Graph Theory Research · graph theory and CDMA systems · Limits and Structures in Graph Theory
