On The Existence Of (3,2,1)-conjugate Orthogonal Latin Squares

Two Latin squares of order v are r-orthogonal if their superposition producesexactly r distinct ordered pairs. If the second square is the (i, j, k)-conjugate of thefrst one, where {i, j, k}={1,2,3}, then the frst square is said to be (i, j, k)-conjugater-orthogonal and denoted by (i, j, k)-r-COLS(v). In this paper, we will investigate theexistence of (3,2,1)-r-COLS(v) and (1,3,2)-r-COLS(v). For v≤49, we provide analmost complete solution for the existence of (3,2,1)-r-COLS(v) with some exceptions.We also show that there exists a (3,2,1)-r-COLS(v) for r∈[v, v~2]\{v+1, v+2,v+3, v+5, v+7, v~23, v~21} with several exceptions if v>49. Since the existenceof (3,2,1)-r-COLS(v) is equivalent to the existence of (1,3,2)-r-COLS(v), we thenhave the same results for (1,3,2)-r-COLS(v).