Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

摘要

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of Z_n such that Δ(x) ∩ Δ(y) = θ holds for any x, y ∈ C, x ≠ y, where Δ(x) = {j — i |i, j ∈ x, i ≠ j}. A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if ∪_(x∈C) Δ(x) =Z_n {0} holds and any codeword x ∈ C has the form {0, i, 2i,..., (k - 1)i}. The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-di fference CACs of weight k = 3 and characterize the code length n 's admitting the condition through a number theoretical approach.