v阶(3,2,1)-共轭r-正交拉丁方在集合r∈{v+2,v+3,v+5}上的不存在性(英文)

Nonexistence of (3,2,1)-conjugate r-orthogonal Latin Squares of Order v for r∈{v+2,v+3,v+5}

2个v阶拉丁方,L=(lij)和M=(mij)被称为是r-正交的,如果把它们重叠起来可以得到恰好r个不同的有序元素偶,即{(lij,mij):1≤i,j≤v}=r,记为r-MOLS(v).r-MOLS(v)在r∈{v+1,v2-1}上的不存在性已经得到证明.如果M是L的(3,2,1)-共轭,可认为L是(3,2,1)-共轭r-正交的,可记为(3,2,1)-r-COLS(v).并且证明了(3,2,1)-r-COLS(v)在r∈{v+2,v+3,v+5}上的不存在性.

Two latin squares of order ν, L = （Iij） and M = （mij） are called to be r-orthogonal if their superposition produces exactly r distinct ordered pairs, that is｜{（lij,mij）：1≤i,j≤ν}｜=r,which is denoted by r-MOLS（ν）. It has been proνed that there does not exist an r-MOLS（ν） for re {ν ＋ 1,ν2 - 1}. If M is the （3,2,1）-conjugate of L, then L is called to be （3,2,1）-conjugate r-orthogonal, as denoted by （3,2,1）-r-COLS（ν）. In this paper, the nonexistence of （3,2,1）-r-COLS（ν） for re {ν＋2,ν＋3,ν＋5} is proved.