Construction of Strongly Mutually Distinct Sudoku Tables and Solid Sudoku Cubes by Cyclotomic Cosets

Abstract

A new method of constructing Sudoku tables (STs) (Sudoku Latin squares) is introduced by making use of individual vectors of cyclotomic cosets of $Z_n$ and their Kronecker product. We show that, it is possible to construct $m$ different STs of order $m$ such that for every $0\leq u, v \leq m-1$ the $(u, v)$ -entry of these $m$ STs is different. These STs could be considered as a perfect set of strongly mutually distinct (SMD) STs, which in turn are used to construct a solid Sudoku cube $(\text{SSC})$ . As a result, a new version of SMD Sudoku puzzles under a new rule and condition is introduced that are interesting for Sudoku puzzles game designers.