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In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius r ≥ 2 and dimension n ≥ 3. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (2011) [5] proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with distance greater than four besides the D P L (3 , 6) code? Later, Horak and AlBdaiwi (2012) [12] conjectured that there are no D P L (n , d) codes for dimension n ≥ 3 and distance d > 4 except for (n , d) = (3 , 6). In this paper, we give a counterexample to this conjecture. Moreover, we prove that for n ≥ 3 , there is a linear D P L (n , 6) code if and only if n = 3 , 11. [ABSTRACT FROM AUTHOR]

Abstract: The necessary conditions for existence of a triplewhist tournament are . By the efforts of many authors through a century, these conditions are shown to be sufficient except for and possibly for . A triplewhist tournament is said to have the three person property if any two games in the tournament do not have three common players. We briefly denote such a design as a . In this paper, we extend the known existence result for s and show that the necessary conditions for existence of a , namely, and , are also sufficient except for and possibly for . [Copyright &y& Elsevier]

Abstract: Frames are useful in dealing with resolvable designs such as resolvable balanced incomplete block designs and triplewhist tournaments. Z-cyclic triplewhist tournament frames are also useful in the constructions of Z-cyclic triplewhist tournaments. In this paper, the concept of an -regular Z-cyclic triplewhist tournament frame is defined, and used to establish several quite general recursive constructions for Z-cyclic triplewhist tournaments. As corollaries, we are able to unify many known constructions for Z-cyclic triplewhist tournaments. As an application, some new Z-cyclic triplewhist tournament frames and Z-cyclic triplewhist tournaments are obtained. The known existence results of such designs are then extended. [Copyright &y& Elsevier]

Abstract: Frequency-hopping multiple-access (FHMA) spread-spectrum communication systems employing multiple frequency shift keying as data modulation technique were investigated by Fuji-Hara, Miao and Mishima [R. Fuji-Hara, Y. Miao, M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inform. Theory 50 (2004) 2408–2420] from a combinatorial approach, where a correspondence between frequency-hopping (FH) sequences and partition-type cyclic difference packings was established, and several combinatorial constructions were provided for FHMA systems with a single optimal FH sequence. In this paper, by means of this correspondence, we describe more combinatorial constructions for such optimal FH sequences. As a consequence, more new infinite series of optimal FH sequences are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we construct CCRRS, complete coupling round robin schedules, for n teams each consisting of two pairs. The motivation for these schedules is a problem in scheduling bridge tournaments. We construct for n a positive integer, , with the possible exceptions of . For n odd, we show that a can be constructed using a house with a special property. For n even, a can be constructed from a Howell design, , with a special property called Property P. We use a combination of direct and recursive constructions to construct with Property P. In order to apply our main recursive construction, we need group divisible designs with odd group sizes and odd block sizes. One of our main results is the existence of these group divisible designs. [Copyright &y& Elsevier]

In this paper, we present a systematic approach to some new direct constructions of block designs. Several examples of non-uniform group divisible designs, resolvable group divisible designs and optimal packings are illustrated to show the advantage of the construction method. [Copyright &y& Elsevier]

The famous Erdős-Falconer distance problems aim to quantify the extent to which large sets must determine many distinct distances. The problems in both the Euclidean setting and the discrete setting have received much attention. An interesting generalization of Erdős-Falconer problems is to find locating point configurations with prescribed metric structure within large subsets. In this paper, we consider such problem in the discrete setting. We say a distance graph H in F q n under Hamming metric means a graph H = (V , E) with vertices as the points in V ⊆ F q n and edges being assigned by Hamming distance between the pairs of vertices. We call a subset A ⊆ F q n contains an isometric copy of distance graph H , if there exists an embedding ϕ : V (H) → A such that for every edge u v ∈ E (H) , the Hamming distance between ϕ (u) and ϕ (v) is equal to the Hamming distance between u and v. When q is a fixed prime power and n goes to infinity, we show that for any given bipartite graph H with ex (n , H) = O (n) , any subset A ⊆ F q n contains a positive proportion of all possible isometric copies of distance graphs H whose edges are assigned by the same Hamming distance, if | A | > q (1 − c) n for some c = c (q , H) > 0. Our methods also work for general bipartite graphs. Moreover, we will give the quantitative bounds for the above constant c = c (q , H). Unlike using analytical method as before, our ideas are more combinatorial, which mainly include two techniques. The first one is the celebrated dependent random choice, which is like glue, sticking several small structures together. The second one is a new extension of the modular version of the classical Delsarte's inequality. In particular, this extension itself is of independent interest, which also helps us build a surprising connection between our main results and the extremal number of the bipartite graphs. [ABSTRACT FROM AUTHOR]