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 an r-MOILS（v） if and only if r∈[v, v^2]／ {v ＋ 1, v^2-1}.