Explicit representation for a class of Type 2 constacyclic codes over the ring ${\mathbb{F}}_{2^m}[u]/⟨u^{2\lambda }⟩$ with even length

Abstract

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $\lambda$ and $k$ be integers satisfying $\lambda ,k\ge 2$ and denote $R={\mathbb{F}}_{2^m}[u]/⟨u^{2\lambda }⟩$. Let $\delta ,\alpha \in {\mathbb{F}}_{2^m}^{\times }$. For any odd positive integer $n$, we give an explicit representation and enumeration for all distinct $(\delta +\alpha u^2)$-constacyclic codes over $R$ of length $2^{k}n$, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every $(\delta +\alpha u^2)$-constacyclic code over $R$ of length $2^{k}n$ is an ideal generated by at most $2$ polynomials in the residue class ring $R[x]/⟨x^{2^{k}n}-(\delta +\alpha u^2)⟩$.

arXiv admin note: text overlap with arXiv:1805.05595