Asymptotically Good Multi-twisted Codes over Finite Chain Rings
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摘要: 对码的渐近性的研究是纠错码理论中的一个核心内容，Shannon第二定理指出，当码长趋于无穷时，存在码率接近信道容量且译码错误概率趋近于零的编码方案。对码的渐近性进行研究可以验证这一理论极限的可达性。在设计和选择编码方案时，渐近性是重要的比较依据，研究码的渐近性有助于理解码的性能极限和设计高效能纠错码，助力实际编码方案的设计与优化，确保其在长码场景下逼近理论最优性能，同时平衡纠错能力、码率与复杂度。该文给出了有限链环上1-生成元多元扭转码是自由码的条件，构造了有限链环上一类自由的1-生成元多元扭转码。基于概率方法和中国剩余定理理论，讨论了这类码的渐近码率和相对距离。结果表明，有限链环上的这类1-生成元多元扭转码是渐近优的。Abstract:
Objective This study aims to address the theoretical gap in the asymptotic analysis of multi-twisted codes over finite chain rings and to provide a foundation for their application in high-efficiency communication and secure data transmission. As modern communication systems demand higher data rates, enhanced error resilience, and robust security, the design of error-correcting codes must balance code rate, error correction capability, and implementation complexity. Finite chain rings, as algebraic structures situated between finite fields and general rings, exhibit a hierarchical ideal structure, that enables sophisticated code designs while retaining the algebraic properties of linear codes. Compared to finite fields, codes over finite chain rings achieve flexible error correction and higher information density through homogeneous weights and Gray mapping. However, existing research has focused primarily on multi-twisted codes over finite fields, leaving the asymptotic properties over finite chain rings unexplored. By constructing 1-generator multi-twisted codes, this work is the first to prove their asymptotic goodness over finite chain rings—i.e., the existence of infinite code sequences$ {\mathcal{C}_i} $with code rate$ R\left( {{\mathcal{C}_i}} \right) $and relative distance$ {{\Delta }}\left( {{\mathcal{C}_i}} \right) $bounded below as code lengths approach infinity. This result not only demonstrates the attainability of Shannon’s Second Theorem in finite chain ring coding but also offers novel solutions for practical systems, such as quantum-resistant encrypted communication and reliable transmission in high-noise channels. Methods In the basic concepts section, the structure of a finite chain ring is defined, utilizing its ideal chain structure to study code generation and properties. The concepts of homogeneous weight are introduced, and the homogeneous distance ${d_{\hom }}$ is established to quantify error correction capabilities. A Gray map is constructed to transform the distance problems over finite chain rings into Hamming distance problems over finite fields. To study the asymptotic properties of multi-twisted codes, 1-generator multi-twisted codes are defined using the module structure of $ {\text{R}}\left[ x \right] $, and their free condition is discussed, as demonstrated in Theorem 1: Each subcode $ {\mathcal{C}_i} = \left\langle {{a_i}\left( x \right)} \right\rangle $ must be a free constant cyclic code, and the rank of $ {\mathcal{C}_i} $ is determined by the degree of the check polynomial$ h(x) $. The asymptotic properties of multi-twisted codes with identical block lengths, which are simpler to analyze than those with varying block lengths are considered. The selection of generators $ ({a_1}(x),{a_2}(x), \ldots ,{a_l}(x)) $ is treated as a random process, defining a probability space. By introducing the ${q^s}$-ary entropy function$ H(x) = x{\log _{{q^s}}}({q^s} - 1) - x{\log _{{q^s}}}x - (1 - x){\log _{{q^s}}}(1 - x) $, the code rate$ R(\mathcal{C}) $and the relative distance $ {{\Delta }}(\mathcal{C}) $ are analyzed. The Chinese Remainder Theorem is applied to decompose the finite chain ring into the direct product of local rings, transforming the global ideal analysis into localized studies to reduce complexity. Finally, it is proven that the relative homogeneous distance and the rate of multi-twisted codes are positively bounded from below. As the code length$ i \to \infty $, the relative distance of the code satisfies $ \Pr \left( {{{\Delta }}\left( {{\mathcal{C}^{\prime (i)}}} \right) \ge \delta } \right) = 1 $(Theorem 2) and $ \Pr \left( {{\text{rank}}\left( {{\mathcal{C}^{\prime (i)}}} \right) = {m_i} - 1} \right) = 1 $(Theorem 3), leading to the conclusion that this class of multi-twisted codes over finite chain rings is asymptotically good. Results and Discussions This paper systematically constructs a class of 1-generator multi-twisted codes (Label 1) over finite chain rings and demonstrates that these codes are asymptotically good based on probabilistic methods and the Chinese Remainder Theorem. This constitutes the first analysis of the asymptotic properties of such codes over finite chain rings. Previous studies on the asymptotic properties of codes have primarily focused on codes over finite fields (e.g., cyclic and quasi-cyclic codes). By leveraging the hierarchical ideal structures of rings (e.g., homogeneous weight and the Chinese Remainder Theorem), the analytical complexity inherent to rings is overcome, thereby extending the scope of asymptotically good codes. This work extends classical finite-field random code analysis to finite chain rings, addressing the complexity of distance computation through complexity via homogeneous weights and Gray mappings. Additionally we leverage the bijection between q-cyclotomic cosets modulo${\text{M}}$ and irreducible factors of ${x^{\text{M}}} - 1$, combined with CRT-based ideal decomposition, significantly simplifies the asymptotic analysis (Lemma 4). Conclusions The asymptotic goodness of multi-twisted codes over finite chain rings has been systematically resoloved, addressing a critical theoretical gap. By constructing 1-generator free codes and applying probabilistic methods combined with the Chinese Remainder Theorem, this work provides the first proof of infinite code sequences over finite chain rings that approach Shannon’s theoretical limits in terms of code rate and relative distance. These codes are suitable for high-frequency communications in 5G/6G networks, deep-space links, and other noisy environments, offering enhanced spectral efficiency through high code rates and robust error correction. This result not only extends the algebraic framework of coding theory but also provides a new coding scheme with strong anti-interference capabilities and high security for practical communication systems. Future research may extend these findings to more complex ring structures and practical application scenarios, further advancing the application of coding theory in the information age. -
Key words:
- Multi-twisted codes /
- Finite chain rings /
- Asymptotic properties
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