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Bierbrauer (2012) developed the theory of q -linear cyclic codes over ( F q ) m and he obtained a parametric description of such codes by cyclotomic cosets. Recently, Cao et al. (2015) obtained the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) , where m is a prime integer. Let R be a finite commutative ring and R n = R [ x ] ∕ 〈 x n − 1 〉 . In this paper, we generalize the theory of F q -linear codes over vector spaces to R -linear codes over free R -algebras (free as R -module). We call these codes, R -additive codes. We introduce a one-to-one correspondence between the classes of cyclic R -additive code and the classes of R n -linear code. Using the structure of R n -linear codes, we present the structure of cyclic R -additive codes, where R is a chain ring. Among other results, q -linear cyclic codes over ( F q ) m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) is given for an arbitrary integer m , not necessarily a prime number. [ABSTRACT FROM AUTHOR]

A dihedral SQS is an SQS admitting a point-regular dihedral automorphism group. Some constructions of dihedral SQSs are established in this paper. We also prove that for v ≡ 2 , 4 ( m o d 6 ) , there is a dihedral SQS ( v ) if there is a dihedral H ( p , 2 , 4 , 3 ) design or an S -cyclic SQS ( 2 p ) for each odd prime divisor p of v . As a corollary, we enlarge the set of known values of v for which there exists a dihedral SQS ( v ) . [ABSTRACT FROM AUTHOR]

Recently, linear codes with a few weights have been constructed and extensively studied. In this paper, for an odd prime p , we determine the complete weight enumerator of two classes of p -ary linear codes constructed from defining set. Our results show that the codes have at most seven weights and may have applications in secret sharing schemes. [ABSTRACT FROM AUTHOR]

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let n be a product of three primes. The problem on the classification of the tetravalent half-arc-transitive graphs of order n has been considered by Xu (1992), Feng et al. (2007) and Wang and Feng (2010), and it was solved for the cases where n is a prime cube or twice a product of two primes. In this paper, we solve this problem for the remaining cases. In particular, there exist some families of these graphs which have a solvable automorphism group but are not metacirculants. [ABSTRACT FROM AUTHOR]

For integer k ≥ 2 and prime power q , an algebraic bipartite graph D ( k , q ) of girth at least k + 4 was introduced by Lazebnik and Ustimenko (1995). Füredi et al. (1995) shown that the girth of D ( k , q ) is equal to k + 5 if k is odd and q is a prime power of form 1 + n ( k + 5 ) / 2 and, conjectured further that D ( k , q ) has girth k + 5 for all odd k and all q ≥ 4 . In this paper, we show that this conjecture is true when ( k + 5 ) / 2 is a power of the characteristic of F q . [ABSTRACT FROM AUTHOR]

Abstract: The paper presents classifications of edge-transitive cyclic regular covers of the complete graphs and , and arc-transitive cyclic regular covers of the complete graph . Two new infinite families of transitive graphs of valency and are found. As an application, tetravalent edge-transitive graphs of order with a prime are classified. [Copyright &y& Elsevier]

Abstract: The finite Figueroa planes are non-Desarguesian projective planes of order for all prime powers , constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundhöfer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Hui and Wong (2012) have shown that these polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unital to be classical, and hence they are not classical. In this article we introduce and make use of a new alternative synthetic description of the Figueroa plane and unital to demonstrate the existence of O’Nan configurations, thus providing support to Piper’s conjecture (1981). [Copyright &y& Elsevier]

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: Suppose is a finite group, such that , where  is prime. We show that if is any generating set of , then there is a Hamiltonian cycle in the corresponding Cayley graph . [Copyright &y& Elsevier]

Abstract: A graph has a representation modulo if there exists an injective map such that vertices and are adjacent if and only if is relatively prime to . The representation number is the smallest such that has a representation modulo . Following earlier work on stars, we study representation numbers of complete bipartite graphs and more generally complete multipartite 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: The present paper is a continuation of the author’s work (Hachenberger (2001) ) on primitivity and complete normality. For certain 2-power extensions over a Galois field , we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of . The main result is as follows: Let and let be the largest integer such that divides ; if , where , then there exists a primitive element in that is completely normal over . Our method not only shows existence but also gives a fairly large lower bound on the number of primitive completely normal elements. In the above case this number is at least . We are further going to discuss lower bounds on the number of such elements in -power extensions, where and , or where is an odd prime, or where is equal to the characteristic of the underlying field. [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: We examine the -ary codes, for any prime , that can be obtained from incidence matrices and line graphs of the Hamming graphs, , obtaining the main parameters of these codes. We show that the codes from the incidence matrices of can be used for full permutation decoding for all . [Copyright &y& Elsevier]

Abstract: In this paper, the authors investigate the question of when a partition of is an -core and also a -core when and are not relatively prime. A characterization of all such -cores is given, as well as a generating function dependent upon the polynomial generating functions for -cores when and are relatively prime. Furthermore, characterizations and generating functions are given for -cores which are self-conjugate and also for -cores. [Copyright &y& Elsevier]

Abstract: A graph is -regular if its automorphism group acts regularly on the set of its -arcs. In this paper, we classify the cubic -regular cubic graphs of order for each and each prime . [Copyright &y& Elsevier]

We show that each non-thin regular association scheme of order p q ( p and q two not necessarily different prime numbers) is a wreath product of two thin cyclic schemes of order p and q . [ABSTRACT FROM AUTHOR]

Abstract: A digraph is -transitive, if for any a path of length , dominates . A digraph is a strong -transitive digraph, if it is -transitive and it is strongly connected. César Hernández-Cruz proposed the following conjecture: Let be a prime and be a strong -transitive digraph. If , contains an -cycle, with and , and is not a symmetrical -cycle, then is a complete digraph. In this paper, we shall prove that the conjecture is true. [Copyright &y& Elsevier]