Dihedral and Quaternion Codes with Orders 8 and 16 Enable Complete Hermitian Dual Code Descriptions for Quantum Codes

The construction of robust and efficient error-correcting codes remains a central challenge in modern information theory, with implications for data storage, communication, and quantum computing. Miguel Sales-Cabrera, Xaro Soler-Escrivà, and Víctor Sotomayor, from Universitat d’Alacant and Universidad de Granada, now present a comprehensive analysis of a specific class of codes built upon dihedral and generalised quaternion groups. Their work delivers a complete algebraic description of these codes’ properties, including their hermitian and euclidean dual structures, achieved through a detailed decomposition of the underlying group algebras. This detailed understanding not only clarifies the mathematical foundations of these codes, but also enables the systematic construction of optimal codes with potential applications in advanced communication systems and, crucially, the development of more resilient quantum codes. Dihedral and Quaternion Codes for Quantum Error Correction This study comprehensively investigates codes constructed from finite groups, with a particular focus on dihedral and generalized quaternion groups, and their potential for correcting errors in quantum computations. Researchers have developed a detailed algebraic description of these codes, providing new insights into their structure and properties. This work builds upon the foundational idea of representing codes as ideals within group algebras, a powerful framework for both code construction and analysis.

The team focused on dihedral and quaternion groups due to their unique algebraic structures and potential for creating effective codes.

This research delivers a complete understanding of the “hermitian dual code” associated with any code linked to a dihedral group, provided the finite field’s characteristic is coprime with the group’s order. This involved utilising the Wedderburn-Artin decomposition of the group algebra, a method that breaks down complex algebras into simpler components. By meticulously computing this decomposition for both dihedral and generalized quaternion groups, the team enabled a detailed analysis of their associated codes. Potential areas for further exploration include investigating the specific parameters of the constructed codes, developing efficient decoding algorithms, and comparing their performance with other known codes. Extending these techniques to other groups could lead to the discovery of new and improved codes, while implementing the code construction and decoding algorithms in software would provide valuable insights into their practical feasibility. Explicitly constructing quantum stabilizer codes from the group code ideals represents a significant step towards realizing the potential of these codes for quantum error correction. This work advances our understanding of algebraic coding theory and its applications to quantum information processing.,.

Group Algebra Decomposition of Dihedral and Quaternion Codes This research pioneers a systematic approach to understanding and constructing linear codes through the lens of group algebra decomposition, specifically focusing on dihedral and generalized quaternion groups. Researchers developed a method to fully describe the hermitian dual code of any code associated with a dihedral group of order 2n, provided the finite field’s characteristic is coprime with 2n. This involved utilising the Wedderburn-Artin decomposition of the group algebra, a technique that breaks down the complex algebra into simpler, more manageable components.

The team meticulously computed this decomposition for both dihedral groups and generalized quaternion groups of order 4n, enabling a detailed analysis of their associated codes. The core of this work lies in applying the Wedderburn-Artin decomposition to the group algebras Fq[Dn] and Fq[Qn], where Fq represents a finite field and Dn and Qn are dihedral and generalized quaternion groups, respectively. This decomposition allows researchers to represent the group algebra as a direct sum of matrix algebras over Fq, providing a clear structural understanding. By leveraging this decomposition, the study fully describes the hermitian dual code for any code linked to these groups, revealing its algebraic properties and structure. Furthermore, the research extends to constructing quantum error-correcting codes, demonstrating the practical application of these theoretical advancements.

The team systematically built these codes using the structure of the group algebra, and notably, successfully rebuilt already known optimal quantum codes using this methodical approach. This confirms the effectiveness of the developed methodology and its potential for generating new and improved quantum codes. This work builds upon previous research, notably the initial introduction of group codes by Berman and MacWilliams, and expands the understanding of non-abelian group codes, which are increasingly relevant in the context of quantum cryptography.,. Dihedral and Quaternion Codes for Quantum Correction This work presents a comprehensive algebraic description of codes derived from finite groups, specifically focusing on dihedral and generalized quaternion groups, and their application to quantum error correction. Scientists achieved a detailed understanding of the “hermitian dual code” associated with any “Dn-code” over a field with 2 elements, building upon the “Wedderburn-Artin’s decomposition” of the relevant group algebra. This decomposition, a cornerstone of the research, allows for a systematic analysis of these codes as ideals within the group algebra structure.

The team refined the existing Wedderburn-Artin decomposition for the group algebra associated with dihedral groups, providing a complete description of the hermitian dual code of any Dn-code whenever the field characteristic does not divide the group order. This refinement significantly reduces the computational complexity compared to previous approaches, as demonstrated by specific examples.

Results demonstrate that all hermitian self-orthogonal Dn-codes can now be fully determined using this new algebraic framework. Further investigations revealed that the semisimple group algebras associated with generalized quaternion groups and dihedral groups are isomorphic, allowing the team to extend their findings to compute the hermitian dual code of any “Qn-code”. Additionally, scientists obtained the euclidean dual code of any Qn-code over a field, a result previously unaddressed in the literature. Based on these hermitian dualities, the team constructed CSS quantum dihedral codes, rebuilding already known optimal quantum error-correcting codes from hermitian self-orthogonal dihedral codes. This methodical approach, leveraging the structure of the group algebra, provides a systematic way to generate these codes, avoiding computationally intensive brute-force methods. The research delivers a powerful new algebraic framework for understanding and constructing quantum error-correcting codes, with potential applications in fault-tolerant quantum computing.,.

Group Algebra Duals Characterised via Decomposition This research presents a comprehensive algebraic description of codes derived from group algebras, specifically focusing on Hermitian and Euclidean duals.

The team successfully characterised the Hermitian dual of any code defined over a finite field using a Wedderburn-Artin decomposition of the relevant group algebra. This allowed for a complete determination of all Hermitian self-orthogonal codes within this framework. Furthermore, the researchers provided a thorough representation of the Euclidean dual code for any code over a generalised quaternion group, again leveraging the Wedderburn-Artin decomposition. Notably, because the group algebras considered share an isomorphism, the description of the Hermitian dual code extends to these related structures. The work extends beyond theoretical characterisation, demonstrating a systematic method for constructing codes via the structure of the group algebra. This approach facilitated the reconstruction of previously known optimal codes, validating the effectiveness of the developed techniques. Future research directions include exploring the application of these techniques to different types of group algebras and investigating the potential for developing more efficient algorithms for code construction and decoding. The findings contribute significantly to the field of coding theory by providing a deeper understanding of the algebraic properties of codes and offering new tools for their design and implementation. 👉 More information 🗞 Dualities of dihedral and generalised quaternion codes and applications to quantum codes 🧠 ArXiv: https://arxiv.org/abs/2512.07354 Tags: