$m$-adic residue codes over $\mathbb{F}_q[v]/(v^s-v)$ and their application to quantum codes
Ferhat Kuruz, Mustafa Sarı, Mehmet E. Koroglu
TL;DR
The paper extends the theory of $m$-adic residue codes to the non-chain ring $\\mathcal{R}_{q,s}=\\mathbb{F}_q[v]/(v^s-v)$ by deriving idempotent generators and dual-containing conditions for codes over this ring. It shows how to realize these codes as direct sums via the ring decomposition with idempotents $\\eta_i$, and uses a preserving-orthogonality Gray map to transfer dual-containing codes to the finite field, enabling CSS-based quantum code constructions. Additionally, the authors obtain optimal or near-optimal code parameters with respect to the Griesmer bound for rings and provide explicit examples of quantum codes arising from Gray images of dual-containing $m$-adic residue codes. Overall, the work broadens quantum-code design to a richer ring setting, offering new families of dual-containing codes and associated quantum codes with practical error-correcting capabilities.
Abstract
Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ and give some examples to illustrate our findings.