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

## TL;DR

This paper provides an explicit representation, enumeration, and structural description of a class of Type 2 constacyclic codes over a specific finite ring, enhancing understanding of their algebraic properties and counting methods.

## Contribution

It introduces a new explicit representation and enumeration formula for $( ext{delta}+ ext{alpha} u^2)$-constacyclic codes over a finite ring of the form $rac{ ext{field}[u]}{ ext{ideal}}$, specifically for odd length codes.

## Key findings

- Explicit formulas for code enumeration.
- Representation of codes as ideals generated by at most two polynomials.
- Complete classification of the codes over the specified ring.

## Abstract

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $\lambda$ and $k$ be integers satisfying $\lambda,k\geq 2$ and denote $R=\mathbb{F}_{2^m}[u]/\langle u^{2\lambda}\rangle$. 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 $2^kn$, 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 $2^kn$ is an ideal generated by at most $2$ polynomials in the residue class ring $R[x]/\langle x^{2^kn}-(\delta+\alpha u^2)\rangle$.