一对正交的不完全拉丁方完备大集

Orthogonal complete large sets of disjoint incomplete Latin squares

不完全拉丁方完备大集,记作LDILS+(n+a,a),由两两不交的n个不完全拉丁方ILS(n+a,a)和a个拉丁方LS(n)构成.正交的不完全拉丁方完备大集,记作OLDILS+(n+a,a),由一对正交的LDILS+(n+a,a)构成.本文研究OLDILS+(n+a,a)的存在性问题,利用有限域上的直接构造以及引入辅助设计OLSn+(n)进行积构造,得到若干OLDILS+(n+a,a)的无穷类.

A complete large set of disjoint ILS(n+a,a)’s,denoted by LDILS+(n+a,a),consists of n ILS(n+a,a)’s and a LS(n)’s,which are pairwise disjoint.An orthogonal complete large set of disjoint ILS(n+a,a)’s,denoted by OLDILS+(n+a,a),consists of a pair of orthogonal LDILS+(n+a,a)’s.In this paper,the existence of OLDILS+(n+a,a) is investigated.We give the direct constructions from the finite field and a product construction by a kind of auxiliary design OLSk+(n) to get some new infinite classes of OLDILS+(n+a,a)’s.