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We show that for every prime number p , a 3-connected non-uniform G F ( p ) -representable matroid can have a clone set of size at most p − 2 . [ABSTRACT FROM AUTHOR]

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]

Let Γ be an abelian group and g ≥ h ≥ 2 be integers. A set A ⊂ Γ is a C h [ g ] -set if given any set X ⊂ Γ with | X | = h , and any set { k 1 , … , k g } ⊂ Γ , at least one of the translates X + k i is not contained in A . For any g ≥ h ≥ 2 , we prove that if A ⊂ { 1 , 2 , … , n } is a C h [ g ] -set in Z , then | A | ≤ ( g − 1 ) 1 / h n 1 − 1 / h + O ( n 1 / 2 − 1 / 2 h ) . We show that for any integer n ≥ 1 , there is a C 3 [ 3 ] -set A ⊂ { 1 , 2 , … , n } with | A | ≥ ( 4 − 2 / 3 + o ( 1 ) ) n 2 / 3 . We also show that for any odd prime p , there is a C 3 [ 3 ] -set A ⊂ F p 3 with | A | ≥ p 2 − p , which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer h ≥ 3 , there is a C h [ h ! + 1 ] -set A ⊂ { 1 , 2 , … , n } with | A | ≥ ( c h + o ( 1 ) ) n 1 − 1 / h . A set A is a weak C h [ g ] -set if we add the condition that the translates X + k 1 , … , X + k g are all pairwise disjoint. We use the probabilistic method to construct weak C h [ g ] -sets in { 1 , 2 , … , n } for any g ≥ h ≥ 2 . Lastly we obtain upper bounds on infinite C h [ g ] -sequences. We prove that for any infinite C h [ g ] -sequence A ⊂ N , we have A ( n ) = O ( n 1 − 1 / h ( log n ) − 1 / h ) for infinitely many n , where A ( n ) = | A ∩ { 1 , 2 , … , n } | . [ABSTRACT FROM AUTHOR]

We consider a problem due to Mariusz Meszka similar to the well-known conjecture of Marco Buratti. Does there exist a near-1-factor in the complete graph on Z p , p is an odd prime, whose set of edge-lengths equals a given multiset L ? We establish several sufficient conditions for the answer to be yes. [ABSTRACT FROM AUTHOR]

Abstract: For every prime power , where , and a prime dividing , we construct a self-orthogonal code and a self-dual code over the field of order . The construction involves Paley graphs and the constructed and codes admit an automorphism group of the Paley graph of order . If is a prime and , where is a non-negative integer, then the self-dual code is equivalent to a Pless symmetry code. In that sense we can view this class of codes as a generalization of Pless symmetry codes. For and we get a self-dual code whose words of minimum weight form a 3-(20, 8, 28) design. [Copyright &y& Elsevier]

Abstract: Given a graph , a subset of is a module of if for each , is adjacent to all the elements of or to none of them. A graph is prime if and the only modules of are , , and singleton vertex sets. Given a prime induced subgraph , we introduce a digraph that yields a necessary and sufficient condition for to be prime. [Copyright &y& Elsevier]

Abstract: In Matula (1968), D.W. Matula described a bijection between and the set of rooted trees; the number is called the Matula number of the rooted tree. The Gutman–Ivić–Matula (GIM) function computes the number of edges of the tree with Matula number . Since there is a prefix-free code for the set of prime numbers such that the codelength of each prime is , we show how some results about the GIM function can be obtained trivially from coding theorems. [Copyright &y& Elsevier]

Abstract: Two sets and of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let and be the counting functions of and . Up to now, all researches on additive complements are under the assumption that and are infinite sets. In this paper, we consider the case that is a finite set. This is very different from the infinite case. [Copyright &y& Elsevier]