New optimized Lcd codes and quantum codes using constacyclic codes over a non-local collection of rings \({{\varvec{A}}}_{{\varvec{k}}}\)

Abstract

In this article, we find several novel and efficient quantum error-correcting codes (\(\boldsymbol{\mathcal{Q}}\)ecc) by studying the structure of constacyclic (\(\boldsymbol{{\mathcal{C}}{\mathcalligra{cc}}}\)), cyclic (\(\boldsymbol{{\mathcal{C}}{\mathcalligra{c}}}\)), and negacyclic codes (N\(\boldsymbol{{\mathcal{C}}{\mathcalligra{c}}}\)) over the ring \({A}_{k}={Z}_{p}\left[{r}_{1},{r}_{2},\dots ,{r}_{k}\right]\)/\(\langle {{(r}_{b}}^{({m}_{b}+1)}-{r}_{b}), {r}_{l}{r}_{b}={r}_{b}{r}_{l}=0, b\ne l\rangle \), where \(p={q}^{m}\) for m, \({m}_{b}\in {\mathbb{N}}\), \({m}_{b} | \left(-1+q\right)\) \(\forall b, l \in \left\{1\, \text{to}\, k\right\}\), \(q\ge 3\) is a prime, \({Z}_{p}\) is a finite field. We define distance-preserving gray map \({\delta }_{k}\). Moreover, we determine the quantum singleton defect (\(\mathcal{Q}\)SD) of \(\boldsymbol{\mathcal{Q}}\)ecc, which indicates their overall quality. We compare our codes with existing codes in recent publications. The rings discussed by Kong et al. (EPJ Quantum Technol 10:1–16, 2023), Suprijanto et al. (Quantum codes constructed from cyclic codes over the ring\(F_{\text{q}}+{\text{vF}}_{\text{q}}+{v}^{2}F_{\text{q}}+{v}^{3}F_{\text{q}}+{v}^{4}F_{\text{q}}\), pp 1–14, 2021. arXiv: 2112.13488v2 [cs.IT]), and Dinh et al. (IEEE Access 8:194082–194091, 2020) are specific cases of our work. Furthermore, we construct several novel and optimum linear complementary dual (Lcd) codes over \({A}_{k}.\)