Optimal conflict-avoiding codes of odd length and weight three

A conflict-avoiding code (CAC) of length n and weight k is a collection of binary vectors of length n and Hamming weight k, such that the inner product of any two vectors or their arbitrary cyclic shifts is at most one. In the study of multiple-access collision channel without feedback, CAC is used to guarantee that each transmitting user can send at least one data packet successfully during a fixed period of time n, provided that at most k users out of M potential users are active at the same time. The number of codewords in a CAC determines the number of potential users in the system. A CAC with maximum cardinality is said to be optimal. In this talk, we focus on the case when n is odd and k = 3, and how to use Graph Theory and Number Theory to characterize the structure of an Optimal CAC.