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Quantum Information Processing

Skew G-codes over $$\mathbb {F}_4[u,v]/\langle u^2, v^2, uv-vu \rangle $$ and new binary entanglement-assisted quantum error-correcting codes

Serap Şahinkaya, Deniz Üstün

Published Jun 27, 2025• 1 citation
Publisher

Abstract

Abstract In this paper, we explore the algebraic structure of skew-G codes over the ring $$A= \mathbb {F}_4[u,v]/\langle u^2, v^2, uv - vu \rangle $$ A = F 4 [ u , v ] / ⟨ u 2 , v 2 , u v - v u ⟩ . In addition to that, we bridge theoretical results with practical applications in quantum computing. More specifically, we derive a significant number of new binary EAQEC codes from the gray images of the skew-G codes over A according to Grassl’s codetable [14].

DOI
10.1007/s11128-025-04796-5

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Skew G-codes over F4[u,v]/⟨u2,v2,uv−vu⟩\mathbb {F}_4[u,v]/\langle u^2, v^2, uv-vu \rangle F4​[u,v]/⟨u2,v2,uv−vu⟩ and new binary entanglement-assisted quantum error-correcting codes
Abstract In this paper, we explore the algebraic structure of skew-G codes over the ring A=F4[u,v]/⟨u2,v2,uv−vu⟩A= \mathbb {F}_4[u,v]/\langle u^2, v^2, uv - vu \rangle A=F4​[u,v]/⟨u2,v2,uv−vu⟩ A = F 4 [ u , v ] / ⟨ u 2 , v 2 , u v - v u ⟩ . In addition to that, we bridge theoretical results with practical applications in quantum computing. More specifically, we derive a significant number of new binary EAQEC codes from the gray images of the skew-G codes over A according to Grassl’s codetable [14].
Maximal entanglement-assisted quantum error correction codes from the skew group ring F4⋊φG{\mathbb {F}}_4 \rtimes _{\varphi } GF4​⋊φ​G by a heuristic search scheme
New non-binary quantum codes from skew constacyclic codes over the ring Fpm+vFpm+v2Fpm\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}Fpm​+vFpm​+v2Fpm​
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.
Abstract In this paper, we study skew constacyclic codes over a class of non-chain rings T=Fq[u1,u2,…,ur]/⟨fi(ui),uiuj−ujui⟩i,j=1r\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^rT=Fq​[u1​,u2​,…,ur​]/⟨fi​(ui​),ui​uj​−uj​ui​⟩i,j=1r​, where q=pmq=p^mq=pm, ppp is some odd prime, mmm is a positive integer and fi(ui)f_i(u_i)fi​(ui​) are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T\mathcal{T}T and their dual. We characterize (Euclidean and Hermitian) dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many Maximum Distance Separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian Linear Complementary Dual (LCD) codes over T\mathcal{T}T, and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous Maximum Distance Separable (MDS) codes and Best-Known Linear Codes (BKLC) over Fq\mathbb{F}_qFq​. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99
Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.
FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes and their application to quantum codes
Let ppp be a prime and Fq\mathbb{F}_qFq​ be the finite field of order q=pmq=p^mq=pm. In this paper, we study FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes where R=Fq+uFq\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_qR=Fq​+uFq​ with u2=uu^2=uu2=u. To characterize FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over FqR\mathbb{F}_q\mathcal{R}Fq​R and obtain their Fq\mathbb{F}_qFq​-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.