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For distributed storage based on data replication, an optimal data placement is vital to minimize the variance of the number of available files. Well-balanced triple systems and nearly well-balanced triple systems (NWBTSs) were proposed to produce optimal data placements for triple replication. This article concentrates on the existence of NWBTSs. By constructing candelabra systems with various desirable partitions, many new constructions for NWBTSs are developed. The main result of this paper is that there always exist optimal data placements for triple replication on a v-set for all positive integers v possibly except when v ≡ 4 (mod 24) or v = 50 , 74 . [ABSTRACT FROM AUTHOR]

An oriented tetrahedron defined on four vertices is a set of four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n with index λ, denoted by TQSλ(n), is a pair (X,ℬ), where X is an n‐set and ℬ is a set of oriented tetrahedra (blocks) such that every cyclic triple on X is contained in exactly λ members of ℬ. A TQSλ(n) is pure if there do not exist two blocks with the same vertex set. When λ=1, the spectrum of a pure TQS(n) has been completely determined by Ji. In this paper, we show that there exists a pure TQS2(n) if and only if n≡1,2(mod 3) and n≥7. A corollary is that a simple QS4(n) also exists if and only if n≡1,2(mod 3) and n≥7. [ABSTRACT FROM AUTHOR]

Abstract: Sylvester’s problem, i.e., the problem of the existence of large sets of Kirkman triple systems (LKTSs), is one of the most challenging problems in combinatorial design theory. Most recursive constructions for LKTSs impose resolvable restrictions on the ingredient designs, making the recursions difficult to apply. In this note we provide necessary small designs by using spherical geometries and a direct construction. Finally, we prove the existence of LKTS for all positive integers . A new result on overlarge sets of Kirkman triple systems is also produced. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: An idempotent Latin square of order is called resolvable and denoted by RILS if the off-diagonal cells can be resolved into disjoint transversals. A large set of resolvable idempotent Latin squares of order , briefly LRILS, is a collection of RILSs pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs. [Copyright &y& Elsevier]

In this article, we mainly consider the existence problem of a group divisible design GDD(3,4,n+s) of type 1ns1. We present two recursive constructions for this configuration using candelabra systems and construct explicitly a few small examples admitting given automorphism groups. As an application, several new infinite classes of GDD(3,4,n+s)s of type 1ns1 are produced. Meanwhile a few new infinite families on candelabra quadruple systems with group sizes being odd and stem size greater than one are also obtained. [ABSTRACT FROM AUTHOR]