Constructions for optimal cyclic ternary constant-weight codes of weight four and distance six

A cyclic ${(n,d,w)}_q$ code is a $q$-ary cyclic code of length $n$, minimum Hamming distance $d$ and weight $w$. In this paper, we investigate cyclic ${(n,6,4)}_3$ codes. A new upper bound on $C{A}_3(n,6,4)$, the largest possible number of codewords in a cyclic ${(n,6,4)}_3$ code, is given. Two new constructions for optimal cyclic ${(n,6,4)}_3$ codes based on cyclic $(n,4,1)$ difference packings are presented. As a consequence, the exact value of $C{A}_3(n,6,4)$ is determined for any positive integer $n\equiv 0,6,18(mod24)$. We also obtain some other infinite classes of optimal cyclic ${(n,6,4)}_3$ codes.