The spectrum for large sets of resolvable idempotent Latin squares

Abstract

An idempotent Latin square of order v $v$ is called resolvable and denoted by RILS(v) if the v(v − 1 ) $v(v-1)$ off‐diagonal cells can be resolved into v − 1 $v-1$ disjoint transversals. A large set of resolvable idempotent Latin squares of order v $v$ , briefly LRILS(v), is a collection of v − 2 $v-2$ RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for v ∈{14 , 20 , 22 , 28 , 34 , 35 , 38 , 40 , 42 , 46 , 50 , 55 , 62 } $v\in \{14,20,22,28,34,35,38,40,42,46,50,55,62\}$ by using multiplier automorphism groups. Hence, there exists an LRILS(v) for any positive integer v ≥ 3 $v\ge 3$ , except v = 6 $v=6$ .