Complete classification of $(\delta +\alpha u^2)$-constacyclic codes over ${\mathbb{F}}_{2^m}[u]/⟨u^4⟩$ of oddly even length

Abstract

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $R={\mathbb{F}}_{2^m}[u]/⟨u^4⟩)$ and $n$ is an odd positive integer. For any $\delta ,\alpha \in {\mathbb{F}}_{2^m}^{\times }$, ideals of the ring $R[x]/⟨x^{2n}-(\delta +\alpha u^2)⟩$ are identified as $(\delta +\alpha u^2)$-constacyclic codes of length $2n$ over $R$. In this paper, an explicit representation and enumeration for all distinct $(\delta +\alpha u^2)$-constacyclic codes of length $2n$ over $R$ are presented.

arXiv admin note: text overlap with arXiv:1511.02369