Junling Zhou

Large sets of Kirkman triple systems and related designs

The problem of the existence of large sets of Kirkman triple systems (LKTS) is one of the most celebrated open problems in design theory. Only a few sparsely distributed infinite classes have been determined, although LKTS have been investigated by many authors. The purpose of this paper is to survey constructions and results on LKTS and related designs. A systematic account of this work is provided. Most of the known constructions are unified and generalized; the approaches to LKTS are enriched; a couple of new constructions for related designs are also displayed. In particular, a new existence class of LKTS is demonstrated and some new results of overlarge sets of Kirkman triple systems are also produced.

Improving Two Recursive Constructions for Covering Arrays

Recursive constructions for covering arrays employ small ingredient covering arrays to build large ones. At present the most effective methods are ‘cut-and-paste’ (or Roux-type) and column replacement techniques. Both can introduce substantial duplication of coverage; if unnecessary duplication can be avoided, then the recursion can yield a smaller array. Two extensions of covering arrays are introduced here for that purpose. The first examines arrays that cover only certain of the t-way interactions; we call these quilting arrays. We develop constructions of such arrays, and generalize column replacement techniques to use them in the construction of covering arrays. The second examines some consequences of nesting covering arrays of smaller strength in those of larger strength; the intersections among the covering arrays so nested lead to improvements in Roux-type constructions. For both directions, we examine consequences for the existence of covering arrays.

The spectrum for large sets of pure Mendelsohn triple systems

An LPMTS(v) is a collection of v-2 disjoint pure Mendelsohn triple systems on the same set of v elements. In this paper, the concept of t-purely partitionable Mendelsohn candelabra system (or t-PPMCS in short) is introduced for constructing LPMTS(v)s. A powerful recursive construction for t-PPMCSs is also displayed by utilizing s-fan designs. Together with direct constructions, the existence of an LPMTS(v) for v=1,9(mod12) and v>1 is established. For odd integer v>=7, a special construction from both LPMTS(v) and OLPMTS(v) to LPMTS(2v+1) is set up. Finally, the existence of an LPMTS(v) is completely determined to be the set {v:v=0,1(mod3),v>=4,v 6,7}.

Direct constructions of large sets of Kirkman triple systems

Research on the existence of large sets of Kirkman triple systems (LKTS) extends from the mid-eighteen hundreds to the present. In this paper we review known direct approaches of constructing LKTS and present new ideas of direct constructions. We finally prove the existence of an LKTS(v) where

Large sets of Kirkman triple systems of prime power sizes

v \in \{69,141,165,213,285,309,333\}

Large sets of resolvable idempotent Latin squares

v∈{69,141,165,213,285,309,333}. Combining recursive constructions yields several new infinite classes.

A pair of disjoint 3-GDDs of type gtu1

Quantum jump codes (JC) are quantum codes which correct errors caused by quantum jumps. A spontaneous emission error design (SEED) has been introduced to construct quantum JC. In this paper the nonexistence of some