Jianguo Lei

Further results on large sets of Kirkman triple systems

An LR design is introduced by the second author in his recent paper and it plays a very important role in the construction of LKTS (a large set of disjoint Kirkman triple system). In this paper, we generalize it and introduce a new design RPICS. Some constructions for these two designs are also presented. With the relationship between them and LKTS, we obtain some new LKTSs.

On large sets of Kirkman systems with holes

Abstract We study the large sets of generalized Kirkman systems. The purpose of introducing the structure is to construct the large sets of Kirkman triple systems (briefly LKTS). Our main result is that there exists an LKTS( v ) for v∈{6·4 n 25 m +3; m,n⩾0} .

On large sets of disjoint Kirkman triple systems

In this paper, we introduce LR(u) designs and use these designs together with large sets of Kirkman triple systems (LKTS) and transitive KTS (TKTS) of order υ to construct an LKTS(uυ). Our main result is that there exists an LKTS(υ) for υ ∈ {3nm(2 . 13k + 1)t; n ≥ 1, k ≥ 1, t = 0,1, m ∈ {1, 5, 11, 17, 25, 35, 43}}.

On large sets of Kirkman triple systems and 3-wise balanced design

Abstract In this paper, the existence of large sets of Kirkman triple system is transformed to the existence of finite OLKFs and LGKSs. Our main result is: If there exist both an OLKF(6k) and an LGKS({3},{6+3},6k−1,6(k−1)+3,3) for all k∈{6,7,…,40}⧹{14,17,21,22,25,26}, then there exists an LKTS(v) for any v≡3 ( mod 6) , v≠21. As well, we present a construction of 3-wise balanced design.

The spectrum for large set of disjoint incomplete Latin squares

Abstract An incomplete Latin square LS(n+a,a) is a Latin square of order n+a with a missing subsquare of order a. A large set of disjoint LS(n+a,a)s, denoted by LDILS(n+a,a), consists of n disjoint LS(n+a,a)s. About the existence of LDILS, Zhu, Wu, Chen and Ge have already obtained some results (see Wu and Zhu, Bull Inst. Combin. Appl., to appear.). In this paper, we introduce a kind of auxiliary design LSm(n) and, using it, completely solve the existence problem of LDILS. The conclusion is that for any positive integer n and any integer a, 0⩽a⩽n , there exists an LDILS(n+a,a) if and only if (n,a)≠(2,1) and (6,5).

On constructions for two dimensional balanced sampling plan excluding contiguous units with block size four

Balanced sampling plans excluding contiguous units (BSEC) were first introduced in 1988 by Hedayat, Rao and Stufken [A.S. Hedayat, C.R. Rao, J. Stufken, Sampling plans excluding contiguous units, J. Statist. Plann. Inference 19 (1988) 159-170]. These designs can be used for survey sampling when the contiguous units provide similar information. In this paper, we show some recursive constructions for two dimensional BSECs with block size four, and give the existence of some infinite classes.

Constructions of large sets of disjoint group-divisible designs LS ( 2 n 4 1 ) using a generalization of * LS ( 2 n )

Large sets of disjoint group-divisible designs with block size three and type 2 n 4 1 were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It is known that such large sets can exist only for n ? 0 ( mod 3 ) and do exist for n = 2 k ( 3 m ) , where m ? 1 ( mod 2 ) and k = 0 , 3 or k ? 5 . A special large set called * LS ( 2 n ) has played a key role in obtaining the above results. In this paper, we shall give a generalization of an * LS ( 2 n ) and use it to obtain a similar result for k = 2 , 4 and partially for k = 1 .

Multimagic rectangles based on large sets of orthogonal arrays

Large sets of orthogonal arrays (LOAs) have been used to construct resilient functions and zigzag functions by Stinson. In this paper, an application of LOAs in constructing multimagic rectangles is given. Further, some recursive constructions for multimagic rectangles are presented, and some infinite families of bimagic rectangles are obtained.