Self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 4n

Abstract

For any odd positive integer n, we express cyclic codes over $${\mathbb {Z}}_4$$ of length 4n in a new way. Based on the expression of each cyclic code $${\mathcal {C}}$$, we provide an efficient encoder and determine the type of $${\mathcal {C}}$$. In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 4n and correct a mistake in the paper “Concatenated structure of cyclic codes over $${\mathbb {Z}}_4$$ of length 4n” (Cao et al. in Appl Algebra Eng Commun Comput 10:279–302, 2016). In addition, we obtain 50 new self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 28.