Encoding and Construction of Quantum Codes from $(γ,Δ)$-cyclic Codes over a Class of Non-chain Rings

TL;DR

The paper addresses constructing high-performance quantum codes from a class of non-chain rings by defining $(\gamma,\Delta)$-cyclic codes over \(\mathscr{R}_{q,s}\) and showing they decompose into $(\theta,\Im)$-cyclic codes over \(\mathbb{F}_q\). It develops dual-containing criteria componentwise, employs a distance-preserving Gray map to translate to classical codes over \(\mathbb{F}_q\), and applies the CSS framework to obtain quantum codes with parameters $[[ (s+1)n, 2k-(s+1)n, d]]_q$. The key contributions include a complete dual-containment analysis for these ring-based codes, a decomposition framework that reduces to well-understood skew-cyclic codes over fields, and numerous new quantum codes with superior parameters compared to existing literature, along with encoding and error-correction procedures suitable for qudits. This work significantly extends quantum-code construction to a broad family of non-chain rings, offering practical pathways to higher-rate quantum codes and new avenues for quantum error correction research.

Abstract

Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(γ,Δ)$-cyclic codes over a class of finite non-chain commutative rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $γ$ is an automorphism of $\mathscr{R}_{q,s}$, $Δ$ is a $γ$-derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(γ,Δ)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(γ,Δ)$-cyclic and $(θ,\Im)$-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.

Theorems & Definitions (41)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.1
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2