The Largest Critical Sets of Latin Squares

Abstract

Latin squares are combinatorial constructions that have found widespread application in communication systems through frequency hopping designs, error correcting codes and encryption algorithms. In this paper, a new, upper bound on the cardinality of the critical sets of all Latin squares of order n is presented. The bound is based on composite group structure and the summatory prime factorisation function (with multiplicities). The new bound aligns with all known, calculated cardinalities of largest critical sets. The proof addresses a long standing, open problem in discrete mathematics and impacts the assurance of systems based on Latin squares. The new bound also reveals a previously unknown, generative relationship between the smallest critical sets scs(n) and the largest critical sets lcs(n) of Latin squares.