Sets of partially orthogonal latin squares and protective planes

摘要

We investigate the construction of sets of t latin squares of a given non-prime-power order q which are as close as possible to being a mutually orthogonal set. The total number of ordered pairs which do not occur when the squares are juxtaposed in pairs will be called the deficiency. Our main result is that complete sets of q - 1 latin squares can be constructed whose deficiency is less than or equal to q(q - 1)~2 when q = 10,12 (that is, it is approximately 18% of the total number of pairs) and sets whose deficiency is less than or equal to 24(q - 1)~2 when q = 14, 15.