Negacyclic codes over the local ring ${\mathbb{Z}}_4[v]/⟨v^2+2v⟩$ of oddly even length and their Gray images

Abstract

Let $R={\mathbb{Z}}_4[v]/⟨v^2+2v⟩={\mathbb{Z}}_4+v{\mathbb{Z}}_4$ ($v^2=2v$) and $n$ be an odd positive integer. Then $R$ is a local non-principal ideal ring of $16$ elements and there is a ${\mathbb{Z}}_4$-linear Gray map from $R$ onto ${\mathbb{Z}}_4^2$ which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over $R$ of length $2n$ are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over $R$ of length $2n$ are presented. Moreover, all $23\cdot (4^p+5\cdot 2^p+9{)}^{\frac{2^p-2}{p}}$ negacyclic codes over $R$ of length $2M_p$ and all $3\cdot (4^p+5\cdot 2^p+9{)}^{\frac{2^{p-1}-1}{p}}$ self-dual codes among them are presented precisely, where $M_p=2^p-1$ is a Mersenne prime. Finally, $36$ new and good self-dual $2$-quasi-twisted linear codes over ${\mathbb{Z}}_4$ with basic parameters $(28,2^{28},d_L=8,d_E=12)$ and of type $2^{14}{4}^7$ and basic parameters $(28,2^{28},d_L=6,d_E=12)$ and of type $2^{16}{4}^6$ which are Gray images of self-dual negacyclic codes over $R$ of length $14$ are listed.

arXiv admin note: text overlap with arXiv:1710.09236