Some Approaches Of Constructing Critical Sets Of Latin Square

Let N be a set with cardinality n and L be a square of order n with entries from N. If every element of N occurs exactly once in each row and column of L, then L is said to be a Latin square over N. If the requirement "exactly" is replaced by "at most then L is said to be a partial Latin square. Further suppose that L is a partial Latin square and satisfies the following two conditions(1) L can be completed to a Latin square in a unique way,(2) removing any element of L destroys that property. Then L is called a critical set or critical latin square, usually denoted by C.Critical set is one of basic objectives in combinatorial design theory and has nice application of cryptography. In1982, Stinson and van Rees [15] studied in depth the problem of constructing critical set. They established a construction approach, called "doubling construction". The present thesis found a more general and effective method of constructing critical set. To be more precise, we proved a conclusion as following:Let C be a h1,h2…,hm-critical set of a latin square L of order h, C be a n1,n2,…,nl-critical set of a latin square L’ of order n. Denote by Chp the hp-critical subset of C containing (i,j; k) and Cnq the nq-critical subset of C containing (ix,jx; kx)(hp∈{h1, h2…, hm}, nq∈{n1, n2…, nl}). If for each ((i,ix),(j,jx);(k,kx)) E C(?)C’, Chp(?)Cnq is a critical set, then C(?)C’ is a critical set of the latin square L*L’...