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

Dec 2022

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 ofconstacyclic codes over chain rings $\mathcal{R}=\frac{\mathbb{Z}_p[u]}{\langleu^2\rangle}$ and $\mathcal{S}=\frac{\mathbb{Z}_p[u]}{\langle u^3\rangle}$. Wealso study additive constacyclic codes over $\mathcal{R}\mathcal{S}$ and$\mathbb{Z}_p\mathcal{R}\mathcal{S}$ using the generator polynomials over therings $\mathcal{R}$ and $\mathcal{S},$ respectively. Further, by defining Graymaps 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 theweight enumeration and MacWilliams identities corresponding to the additivecodes over $\mathbb{Z}_p\mathcal{R}\mathcal{S}$. Finally, as an application ofthe obtained codes, we give quantum codes using the CSS construction.

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