Abstract
In this paper, we consider optimal $q$ -ary cyclic constant-weight codes of length $n$ , minimum distance $d$ , and weight $w$ , briefly cyclic $(n,d,w)_{q}$ codes. We introduce the pure and mixed difference method to present a combinatorial description for a cyclic $(n,d,w)_{q}$ code and then obtain some tight upper bounds on the sizes of optimal cyclic $(n,d,w)_{q}$ codes. Finally, by using Skolem-type sequences, we completely determine the sizes of optimal cyclic $(n,d,3)_{3}$ codes with minimum distance $1\leq d\leq 6$ .
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