New optimal constructions of conflict-avoiding codes of odd length and weight 3

Conflict-avoiding codes (CACs) have played an important role in multiple-access collision channel without feedback. The size of a CAC is the number of codewords which equals the number of potential users that can be supported in the system. A CAC with maximal code size is said to be optimal. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, the maximal sizes of both equidifference and non equidifference CACs of odd prime length and weight 3 are obtained. Meanwhile, the optimal constructions of both equidifference and non equidifference CACs are presented. The numbers of equidifference and non equidifference codewords in an optimal code are also obtained. Furthermore, a new modified recursive construction of CACs for any odd length is shown. Non equidifference codes can be constructed in this method.