An Investigation of \(2\)-Critical Sets in Latin Squares

Abstract

In this paper, we focus on the existence of \(2\)-critical sets in the latin square corresponding to the elementary abelian \(2\)-group of order \(2^n\). It has been shown by Stinson and van Rees that this latin square contains a \(2\)-critical set of volume \(4^n – 3^n\). We provide constructions for \(2\)-critical sets containing \(4^n – 3^n + 1 – \left(2^{k-1} + 2^{m-1} + 2^{n-(k+m+1)}\right)\) entries, where \(1 \leq k \leq n\) and \(1 \leq m \leq n – k\). That is, we construct \(2\)-critical sets for certain values less than \(4^n – 3^n + 1 – 3\cdot 2^{\lfloor n/3\rfloor – 1}\). The results raise the interesting question of whether, for the given latin square, it is possible to construct \(2\)-critical sets of volume \(m\), where \(4^n – 3^n + 1 – 3\cdot 2^{\lfloor n/3\rfloor – 1} < m < 4^n – 3^n\).