Additive constacyclic codes and the MacWilliams identities over mixed alphabets
DOI:
https://doi.org/10.13069/jacodesmath.v12i2.344Keywords:
Constacyclic code, Chain ring, Frobenius ring, Gray image, MacWilliams identityAbstract
Let $\mathbb{Z}_p$ be the ring of integers modulo a prime integer $p$, 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 provide the weight enumeration and MacWilliams identities corresponding to the additive codes over $\mathbb{Z}_p\mathcal{R}\mathcal{S}$.
Received: 17 August 2024 | Accepted: 18 January 2025Downloads
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