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Let F q be a finite field with q elements, l an odd prime, and t , v positive integers such that l v | ( q − 1 ) and gcd ( t , l ) = 1 . In this paper, we give a combinatorial result and use it to determine the weight distribution of a cyclic code of length t l m with t | ( q − 1 ) , which is an open question in Yang et al. (2013). Moreover we compute the weight distributions of cyclic codes of length t l m , where q ≡ 3 ( mod 4 ) and t = 4 or 8 . [ABSTRACT FROM AUTHOR]

Abstract: In this article a complete classification of tetravalent arc-transitive graphs of order twice a product of two primes is given. [Copyright &y& Elsevier]

Abstract: A graph is said to be -prime if, whenever it is a subgraph of a nontrivial Cartesian product graph, it is a subgraph of one of the factors. A diagonalized Cartesian product is obtained from a Cartesian product graph by connecting two vertices of maximal distance by an additional edge. We show there that a diagonalized product of -prime graphs is again -prime. Klavžar et al. [S. Klavžar, A. Lipovec, M. Petkovšek, On subgraphs of Cartesian product graphs, Discrete Math. 244 (2002) 223–230] proved that a graph is -prime if and only if it admits a nontrivial path--coloring. We derive here a characterization of all path--colorings of Cartesian products of -prime graphs. [Copyright &y& Elsevier]

Abstract: Let be a Hadamard -design. Suppose that the prime divides , but that does not divide . A result of Klemm implies that every residual design of has -rank at least . Also, every derived design of has -rank at least if . We show that when is a skew Hadamard design, the -ranks of the residual and derived designs are at least even if divides or . We construct infinitely many examples where the -rank is exactly . [Copyright &y& Elsevier]

Abstract: Let denote the number of times the prime number appears in the prime factorization of the integer . The following result is proved: If there is a perfect 1-error correcting code of length  over an alphabet with  symbols then, for every prime number . This condition is stronger than both the packing condition and the necessary condition given by the Lloyd theorem, as it for example excludes the existence of a perfect code with the parameters . [Copyright &y& Elsevier]

Abstract: We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups and with a suitable power of a prime number . [Copyright &y& Elsevier]

Abstract: Let be a finite abelian group and let be an integer. A sequence of elements in is called a -barycentric sequence if there exists such that . The -barycentric Davenport constant is defined to be the smallest number such that every sequence in of length contains a -barycentric subsequence. In this paper, we prove that if is a prime, then for , which improves a result of Delorme et al. [ABSTRACT FROM AUTHOR]

Abstract: A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. By Folkman [J. Folkman, Regular line-symmetric graphs, J. Combin Theory 3 (1967) 215–232], there is no semisymmetric graph of order or for a prime and by Malnič, et al. [A. Malnič, D. Marušič, C.Q. Wang, Cubic edge-transitive graphs of order , Discrete Math. 274 (2004) 187–198], there exists a unique cubic semisymmetric graph of order , the so-called Gray graph of order 54. In this paper it is shown that a connected cubic semisymmetric graph of order exists if and only if is divisible by 3. There are exactly two such graphs for a given order, which are constructed explicitly. [Copyright &y& Elsevier]

Abstract: There are well-known necessary conditions for the existence of a generalized Bhaskar Rao design over a group , with block size . It has been conjectured that these necessary conditions are indeed sufficient. We prove that they are sufficient for groups of order where are primes and for groups of all orders except possibly 32, 36, 48, 54, 60, 64, 72, 96. [Copyright &y& Elsevier]

Abstract: Write and where is an odd prime. Let be a value that is congruent () to either or . For any from , evaluate each of and within the interval . Then consider the quantity where the differences are evaluated in the interval , and the quantity where the differences are evaluated () in the interval . As varies over , the values of each of and give exactly two occurrences of nearly every member of . This fact enables and to be used in constructing some terraces for and from segments of elements that are themselves initially evaluated in . [Copyright &y& Elsevier]

Abstract: We prove a theorem that for an integer , if is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of , where , is at least . By some number-theoretic arguments there are an infinite number of integers s satisfying the hypothesis of the theorem. The theorem is the first known example of constructing at least , , nonisomorphic nonorientable triangular embeddings of for , . To prove the theorem, we use a new approach to constructing nonisomorphic triangular embeddings of complete graphs. The approach combines a cut-and-paste technique and the index one current graph technique. A new connection between Steiner triple systems and constructing triangular embeddings of complete graphs is given. [Copyright &y& Elsevier]

Abstract: The group is 3-homogeneous on the projective line when q is a prime power congruent to 3 modulo 4 and therefore it can be used to construct 3-designs. In this paper, we determine all 3-designs admitting with block size not congruent to 0 and 1 modulo p where . [Copyright &y& Elsevier]

Abstract: We show that for each integer n for which there is a Hadamard matrix of order and is a prime number, there is a productive regular Hadamard matrix of order . As a corollary, by applying a recent result of Ionin, we get many parametrically new classes of symmetric designs whenever either of or is a prime power. [Copyright &y& Elsevier]

Abstract: We prove a conjecture of Spiga by showing that the number of nonisomorphic Cayley digraphs of , a prime, that are not CI-digraphs of approaches infinity as approaches infinity. [Copyright &y& Elsevier]