Bounds and Constructions for Optimal Constant Weight Conflict-Avoiding Codes

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 Sigma_{0 les t les n-1} x_it x_j, _t+s les lambda for any distinct codewords x_j = (x_i0,x_i1,hellip,x_{i, n-1}) and x_j = (x_j0, x_j1,hellip, x_{j, 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.