Optimal tight equi-difference conflict-avoiding codes of length n = 2 ^k ± 1 and weight 3

For a k-subset X of Zn, the set of differences on X is the set ΔX={i-j (mod n): i,jεX,i≠j}. A conflict-avoiding code CAC of length n and weight k is a collection C of k-subsets of Zn such that ΔXΔY = ø for any distinct X,YεC. Let CAC(n,k) be the class of all the CACs of length n and weight k. The maximum size of codes in CAC(n, k) is denoted by M(n,k). A code Cε CAC(n, k) is said to be optimal if |C| = M(n,k). An optimal code C is tight equi-difference if |CΔX = Zn\{0} and each codeword in C is of the form {0,i,2i,⋯,(k-1)i}. In this paper, the necessary and sufficient conditions for the existence problem of optimal tight equi-difference conflict-avoiding codes of length n = 2k±1 and weight 3 are given.