Existence of (3, 2, 1)-conjugate r-orthogonal Latin squares

Abstract

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 nonexistence of (3, 2, 1)-r-COLS(v )f orr ∈{ 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).