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We give a direct construction of a (Zp×G,{0}×G,4,1) relative difference family for G∈{Z6×Z6,Z2×Z18,Z6×Z18,Z2×Z54} and every prime p≡3(mod4) with p>3. These allow us to construct an optimal (Z6m×Z6n,4,1) difference packing and an optimal (Z2m×Z18n,4,1) difference packing for every pair of positive integers (m,n). The corresponding optimal optical orthogonal signature pattern codes are also obtained. [ABSTRACT FROM AUTHOR]

An H(m,g,k,3) design is a triple (X,T,B), where X is a set of mg points, T a partition of X into m disjoint sets of size g, and B a set of k‐element transverses of T, such that each 3‐element transverse of T is contained in exactly one of them. In 1990, Mills determined the existence of an H(m,g,4,3) design with m≠5. In this paper, an efficient construction shows that an H(5,g,4,3) exists for any integer g≡2,10(mod12) with g≥10. Consequently, the necessary and sufficient conditions for the existence of an H(m,g,4,3) design are m≥4, mg≡0(mod2), and g(m−1)(m−2)(mod3), with a definite exception (m,g)=(5,2). [ABSTRACT FROM AUTHOR]

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Candelabra quadruple systems (CQS) were first introduced by Hanani who used them to determine the existence of Steiner quadruple systems. In this paper, a new method has been developed by constructing partial candelabra quadruple systems with odd group size, which is a generalization of the even cases, to complete a design. New results of candelabra quadruple systems have been obtained, i.e. we show that for any [ABSTRACT FROM AUTHOR]

Symmetric orthogonal arrays and mixed orthogonal arrays are useful in the design of various experiments. They are also a fundamental tool in the construction of various combinatorial configurations. In this paper, we investigated the mixed orthogonal arrays with four and five factors of strength two, and proved that the necessary conditions of such mixed orthogonal arrays are also sufficient with several exceptions and one possible exception. [ABSTRACT FROM AUTHOR]