Quantum Codes from additive constacyclic codes over a mixed alphabet and the MacWilliams identities

TL;DR

This paper explores the construction of quantum codes from additive constacyclic codes over specific chain rings and their Gray images, establishing MacWilliams identities and weight enumerations for these codes.

Contribution

It introduces new additive constacyclic codes over mixed rings and derives their MacWilliams identities, enabling the construction of quantum codes via the CSS method.

Findings

Derived weight enumerations for additive codes.

Established MacWilliams identities for these codes.

Constructed quantum codes using CSS from the new codes.

Abstract

Let $\mathbb{Z}_p$ be the ring of integers modulo a prime number $p$ where $p-1$ is a quadratic residue modulo $p$. This paper presents the study of constacyclic codes over chain rings $\mathcal{R}=\frac{\mathbb{Z}_p[u]}{\langle u^2\rangle}$ and $\mathcal{S}=\frac{\mathbb{Z}_p[u]}{\langle u^3\rangle}$. We also study additive constacyclic codes over $\mathcal{R}\mathcal{S}$ and $\mathbb{Z}_p\mathcal{R}\mathcal{S}$ using the generator polynomials over the rings $\mathcal{R}$ and $\mathcal{S},$ respectively. Further, by defining Gray maps on $\mathcal{R}$, $\mathcal{S}$ and $\mathbb{Z}_p\mathcal{R}\mathcal{S},$ we obtain some results on the Gray images of additive codes. Then we give the weight enumeration and MacWilliams identities corresponding to the additive codes over $\mathbb{Z}_p\mathcal{R}\mathcal{S}$. Finally, as an application of the obtained codes, we give quantum codes using the CSS construction.