Optimal equi-difference conflict-avoiding codes of weight four

A conflict-avoiding code (CAC) of length $$n$$n and weight $$w$$w is defined as a family $${\mathcal C}$$C of $$w$$w-subsets (called codewords) of $${\mathbb {Z}}_n$$Zn, the ring of residues modulo $$n$$n, such that $$\Delta (C) \cap \Delta (C') = \emptyset $$Δ(C)∩Δ(C′)=∅ for any $$C, C' \in {\mathcal C}$$C,C′∈C, where $$\Delta (C) = \{ j-i \pmod {n} : i, j \in C, i \ne j\}$$Δ(C)={j-i(modn):i,j∈C,i≠j}. A code $${\mathcal C}$$C in CACs of length $$n$$n and weight $$w$$w is called an equi-difference code if every codeword $$C \in {\mathcal C}$$C∈C has the form $$\{ 0, i, 2i, \ldots , (w-1) i \}$${0,i,2i,…,(w-1)i}. A code $${\mathcal C}$$C in CACs of length $$n$$n and weight $$w$$w is said to be optimal if $${\mathcal C}$$C has the maximum number of codewords. In this article, we investigate sizes and constructions of optimal codes in equi-difference CACs of weight four by using properly defined directed graphs. As a consequence, several series of infinite number of optimal equi-difference CACs are also provided.