The Smallest 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, lower bound on the cardinality of the critical sets of all Latin squares of order n is shown to be Ω(n!(n-3)!) where Ω is the summatory prime factorisation function (with multiplicities). The proof draws on the direct product of the symmetric groups Sn and Sn-3. The smallest critical set cardinalities of the new bound align with its known, calculated values and reduce previously proven bounds for n > 8. The proof refutes the long standing Nelder conjecture.