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Abstract The combinatorial object named t -spontaneous emission error design (t -SEED) was proposed by Beth et al. in 2003 to correct errors caused by quantum jumps. The newly rising category of t -SEEDs contains the classical problem of large sets as a subbranch at the extremal optimal situation. In this article we study carefully the necessary conditions for the existence of a 3- (v , 4 ; m) SEED where v ≡ 5 (mod 6), v ≥ 23 and m = v − 7 (the largest possible dimension). By considering the derived designs we determine all possible degeneracy graphs to twenty types (including the empty graph). Then we define triad graphs to investigate these degeneracy graphs. Finally we rule out eighteen of them and reduce the leave of the putative optimal 3-SEED to three concrete types. [ABSTRACT FROM AUTHOR]

A t-spontaneous emission error design, denoted by t- [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: 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]

Abstract: The necessary and sufficient conditions for the existence of a -GDD of type have been established by Colbourn et al. In this paper, the existence of a -GDD of type for any is investigated. Finally, its existence spectrum is completely determined. [Copyright &y& Elsevier]