On the Spectrum of Mutually r orthogonal Idempotent Latin Squares

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. The two squares are said to be r-orthogonal idempotent Latin squares and denoted by r-MOILS(v) if they are all idempotent. In this paper, we show that for any integer v >= 28, there exists