Constructing self-dual cyclic codes over $${\mathbb {Z}}_{9}$$ Z 9 of length 3n

Abstract

In this paper, we study cyclic codes over $$\mathbb {Z}_9$$ of length 3n, where n is a positive integer satisfying $$\mathrm{gcd}(3,n)=1$$ . First, a canonical form decomposition of any cyclic code over $$\mathbb {Z}_9$$ of length 3n are given and a unique set of generators for each subcode is presented. Hence the structure of any cyclic code over $$\mathbb {Z}_9$$ of length 3n is determined. From this decomposition, formulas for the number of all codes and the number of codewords in each code are given. Then dual codes and self-duality of these codes are investigated. As an application, all 10061824 distinct cyclic codes over $$\mathbb {Z}_9$$ of length 24 and all 544 self-dual codes among them are listed explicitly. Moreover, 280 new and good self-dual cyclic codes over $$\mathbb {Z}_9$$ with basic parameters $$\left( 24, 3^{24}, 3\right) $$ are obtained.