Nonexistence of (3, 2, 1)-conjugate (v+7)-orthogonal Latin squares

摘要

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the (3, 2, 1)-conjugate of the first one, we say that the first square is (3, 2, 1)- conjugate r-orthogonal, denoted by (3, 2, 1)-r-COLS(v). The nonexis-tence of (3, 2, 1)-r-COLS(v) for r ∈ {v + 2, v + 3, v + 5} has been proved by Zhang and Xu [Int. J. Combin. Graph Theory Applic. 2 no. 2 (2009), 103-109]. In this paper, we show the nonexistence of (3, 2, 1)-(v + 7)- COLS(v).