关于FSOLS(2~nu^m)的存在性研究

On Existence of FSOLS of Type 2~nu^m

如果一个v阶自正交拉丁方(SOLS)有in个阶为ih的子-SOLS(1≤i≤k),它们互不相交且是生成的,即∑k i=l nihi=v,就称这个自正交拉丁方为frame SOLS,记作FSOLS(h1n1h2n2…hknk).本文讨论FSOLS(2num)(m≥3,u为偶数)的存在性问题,主要利用了填洞构造法和加权构造法,得到FSOLS(2num)的存在条件如下:(1)m=3,u=4,n≥22;u=6,n≥31;u≥8,n≥u/2且n≠u/2+2,u/2+3;(2)m≥4,u≥8,n≥4.

An SOLS （self orthogonal Latin square） of order v with ni missing sub-SOLS （holes） of order hi （1 ≤i≤ k）, which are disjoint and spanning （i.e.∑i=1^knihi=v）, is called a frame SOLS and denoted by FSOLS （h1^n1h2^n2…hk^nk）. In this paper the author discusses the existence of FSOLS（2^nu^m） for m ≥3 and u is even, in which the filling in holes and weighting constructions are mainly used. It may conclude with the following results： （1） m=3, u=4, n≥22;u=6, n≥31;u≥8, n≥u/2 and n≠u/2＋2, u/2＋3. （2） m≥ 4, u ≥8, n ≥ 4.