Complete classification for simple root cyclic codes over the local ring $\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle $

Abstract

Let $R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle $. Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over $\mathbb {Z}_{4}$ and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over $\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle $.