(3,2,1)-共轭r-正交拉丁方的存在性

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

如果两个v阶拉丁方L和M的重叠产生恰好r个不同的有序对,则称L和M是r-正交的.如果L还是M的(i,j,k)-共轭,则称L是(i,j,k)-共轭r-正交的,简记为(i,j,k)-r-COLS(v)((i,j,k)-r-conjugate orthogonal Latin square of order v),其中{i,j,k}={1,2,3}.本文研究(3,2,1)-r-COLS(v)的存在性问题.对于v 23,除去少数几个可能的例外值,本文给出关于(3,2,1)-r-COLS(v)的几乎完整的解.对于v>23,如果r∈[v,v2]\{v+1,v+2,v+3,v+5,v+7,v2 1},除去可能的例外r=v2 3,都存在(3,2,1)-r-COLS(v).由于(3,2,1)-r-COLS(v)的存在性与(1,3,2)-r-COLS(v)的存在性是等价的,本文得到关于(1,3,2)-r-COLS(v)的同样结论.

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the （i,j, k）-conjugate of the first one, where {i,j, k} = {1, 2, 3}, then the first square is said to be （i, j, k）-conjugate r-orthogonal and denoted by （i, j, k）-r-COLS（v）. We will investigate the existence of （3,2, 1）-r-COLS（v）. For v ~〈 23, we provide an almost complete solution for the existence of （3, 2, 1）-r-COLS（v） with some possible exceptions. For v 〉 23, there is a （3, 2, 1）-r-COLS（v） if r E Iv, v2]／ {v ＋ 1, v ＋ 2, v ＋ 3, v ＋ 5, v ＋ 7, v2 - 1} except possibly when r = v2 - 3. Since the existence of （3, 2, 1）-r-COLS（v） is equivalent to the existence of （1, 3, 2）-r-COLS（v）, we then have the same results for （1, 3, 2）-r-COLS（v）.