We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic classes in finite fields.

The duals of cyclic codes with two zeros have been extensively studied, and their weight distributions have recently been evaluated in some cases. In this paper, we determine the weight distribution of a certain new class of such codes by computations involving index 2 Gauss sums.

In this paper, a tight upper bound on the maximum possible code size of n, 4, 2, 1)-OOCs and some direct and recursive constructions of optimal (n, 4, 2, 1)-OOCs attaining the upper bound are given. As consequences, the following new infinite series of optimal (gn,4,2,1)-OOCs are obtained: i) g isin {1,7,11,19,23,31,35,59,71,79,131,179,191,239,251,271,311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971} or g is a primeh25i49ip1p2hellipprwhere h isin {0,1}, i and j are arbitrary nonnegative integers, and each piis a prime equiv 1 ( mod 8); ii) g = 2g' where g' isin {1,7,11,19,23,31,47,71,127,151,167,191,263,271,311,359,367,383,431,439,463,479,503,631,647,719,727,743,823,839,863,887,911,919,967,983,991} and n = p1p2hellipprwhere each piis a prime equiv 1 ( mod 4); iii) g isin {4,20} and n is any positive integer prime to 30; iv) g = 8 and n= p1p2hellipprwhere each piis a primary equiv 1 ( mod 4) greater than 5.

A conflict-avoiding code (CAC) C of length n with weight k is a family of binary sequences of length n and weight k satisfying Sigma0lestlesn-1xitxj,t+sles lambda for any distinct codewords xj= (xi0,xi1,hellip,xi,n-1) and xj= (xj0, xj1,hellip, xj,n-1) in C and for any integer s, where the subscripts are taken modulo n. A CAC with maximal code size for given n and k is said to be optimal. A CAC has been studied for sending messages correctly through a multiple-access channel. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, the case lambda = 1 is treated, and various direct and recursive constructions of optimal CACs for weight k = 4 and 5 are obtained by providing constructions of CACs for general weight k. In particular, the maximum code size of CACs satisfying certain sufficient conditions is determined through number theoretical and combinatorial approaches.

A conflict-avoiding code (CAC) C of length n with weight k is a family of binary sequences of length n and weight k satisfying $\sum_{0\le t\le n-1}x_{it}x_{j,t+s}\le \lambda$ for any distinct codewords $x_i=(x_{i0},x_{i1},\ldots,x_{i,n-1})$ and $x_j=(x_{j0},x_{j1},\ldots,x_{j,n-1})$ in C and for any integer s, where the subscripts are taken modulo n. A CAC with maximum code size for given n and k is said to be optimal. A CAC has been studied for sending messages correctly through a multiple-access channel. The use of an optimal CAC enables the largest possible number of potential users to transmit information efficiently and reliably. In this paper, the case $\lambda=1$ is treated, and various direct and recursive constructions of optimal CACs for weight $k=4$ and 5 are obtained by providing constructions of CACs for general weight k. In particular, the maximum code size of CACs satisfying certain sufficient conditions is determined through number theoretical and combinatorial approaches.