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

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 $\mathcal{R}=\frac{{\mathbb{Z}}_p[u]}{⟨u^2⟩}$ and $\mathcal{S}=\frac{{\mathbb{Z}}_p[u]}{⟨u^3⟩}$. 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.

arXiv comment22 pages