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The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem of constructing optimal ternary constant-composition codes with Hamming weight four and minimum distance six. The problem is solved with a small number of lengths undetermined. The previously known results are those with code length no greater than 10. [ABSTRACT FROM AUTHOR]

Abstract: A graph is an -graph if its vertices can be labeled by binary vectors in such a way that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding vertices. In this note, we study when the result of gluing two -graphs along an edge is an -graph. If at least one of the constituent graphs is bipartite, then the resulting graph is also an -graph. For two nonbipartite -graphs, this is not always the case. [Copyright &y& Elsevier]

In this paper 1 1 An extended abstract of this paper appeared in Proceedings of 11th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2017) (Glazik et al. 2017) [8]. we introduce a variant of Mastermind, which we call Static Permutation Mastermind and which is defined as follows. One player creates a secret code, called secret, consisting of n colors on n pegs, where each color is used exactly once. The second player tries to determine the secret with as few questions as possible, where each question is a possible secret and all questions are asked at once. After having asked all questions, he receives for each question the number of correctly placed colors and has one more try to determine the secret. The main result of this paper is an upper bound of O (n 1. 525) questions. It is proved by a distinction of pairs of possible secrets with low and high Hamming distance. For pairs with a low Hamming distance we construct questions using certain arithmetic progressions. For pairs with a high Hamming distance we estimate the size of a vertex cover in a suitable hypergraph. With a slight modification of the arguments of Doerr et al. (2016) we also give a lower bound of Ω (n log n). [ABSTRACT FROM AUTHOR]