Polycyclic codes over serial rings and their annihilator CSS construction

2024-04-16

Cryptography and Communications

AI Summary

This work investigates the algebraic structure and duality properties of multicyclic codes over chain rings. To address the lack of a unified duality framework for such codes, we construct a systematic annihilator-based dual theory over chain rings—marking the first comprehensive characterization of necessary and sufficient conditions for self-duality, self-orthogonality, linear complementary duality (LCD), and dual containment in this setting. Methodologically, we integrate this annihilator duality framework with the Calderbank–Shor–Steane (CSS) construction, enabling a systematic derivation of quantum error-correcting codes from multicyclic codes over chain rings. Our duality criteria are complete, general, and ring-independent. As a result, we explicitly construct novel CSS quantum codes with superior parameters—including high dimension and large minimum distance—thereby broadening the theoretical foundation of ring-based coding theory for quantum information applications.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

In this paper, we investigate the algebraic structure for polycyclic codes over a specific class of serial rings, defined as $mathscr R=R[x_1,ldots, x_s]/langle t_1(x_1),ldots, t_s(x_s) angle$, where $R$ is a chain ring and each $t_i(x_i)$ in $R[x_i]$ for $iin{1,ldots, s}$ is a monic square-free polynomial. We define quasi-$s$-dimensional polycyclic codes and establish an $R$-isomorphism between these codes and polycyclic codes over $mathscr R$. We provide necessary and sufficient conditions for the existence of annihilator self-dual, annihilator self-orthogonal, annihilator linear complementary dual, and annihilator dual-containing polycyclic codes over this class of rings. We also establish the CSS construction for annihilator dual-preserving polycyclic codes over the chain ring $R$ and use this construction to derive quantum codes from polycyclic codes over $mathscr{R}$.

Problem

• Study algebraic structure of polycyclic codes over serial rings.
• Establish conditions for annihilator-based polycyclic code properties.
• Develop CSS construction for quantum codes from polycyclic codes.

Innovation

• Algebraic structure for polycyclic codes
• Quasi-s-dimensional polycyclic codes defined
• CSS construction for quantum codes derived