Self-orthogonal Quasi-Cyclic Codes Constructed Using Constituent Codes Enable Error-Correcting Codes with Improved Parameters

Error-correcting codes are vital for reliable data transmission, and recent advances focus on quasi-cyclic codes as a promising approach to building more efficient systems. Gustavo Terra Bastos, Angelynn Álvarez, and Cameron Williams present a new method for creating an unlimited number of these codes, specifically designed to be self-orthogonal under both Euclidean and Hermitian calculations. This construction leverages the properties of smaller, foundational codes, allowing the researchers to determine the codes’ dimensions and establish a strong lower limit for their error-correcting capability. By demonstrating how these codes achieve performance approaching theoretical limits and can generate self-dual variations, this work significantly expands the toolkit for designing robust and high-performing communication systems.

The minimum distance is computed using constituent codes defined over field extensions of finite fields. This research also demonstrates that the lower bound for the minimum distance satisfies a square-root-like relationship and illustrates how self-dual quasi-cyclic codes can arise from this construction. Utilizing the CSS construction, the existence of quantum error-correcting codes with good parameters is shown, essential for reliable quantum information processing and increasingly demanded with advancements in quantum computation and communication.

Constructing Self-Orthogonal Quasi-Cyclic Codes

Scientists have developed a systematic method for constructing infinite families of quasi-cyclic codes possessing self-orthogonal and self-dual properties, crucial for advancing quantum error correction. The research centers on leveraging constituent codes, a technique allowing for the systematic design of these complex codes, and applies this approach to both Euclidean and Hermitian inner products. This construction begins with defining codes over finite fields, focusing on codes where the number of elements is a prime number raised to a power, and engineers a process for determining the dimension and a lower bound for the minimum distance of these codes. To extend the construction to codes self-orthogonal with respect to the Hermitian inner product, scientists introduced the concept of Galois-closed codes, broadening the applicability of their method. The study pioneered a technique for constructing quantum error-correcting codes with improved parameters by utilizing the CSS construction, demonstrating that codes generated through this method yield quantum error-correcting codes with parameters exceeding previously known limits.

Quasi-Cyclic MDS Codes for Quantum Error Correction

This research investigates quasi-cyclic codes as a promising approach to constructing quantum error-correcting codes, focusing on leveraging the properties of Maximum Distance Separable (MDS) codes, which are optimal in terms of error-correcting capability. Scientists explore the use of self-orthogonal and Hermitian self-orthogonal codes as building blocks for quantum codes, presenting new methods for constructing quantum MDS codes, highly desirable due to their optimal error-correcting capabilities. The research leverages the algebraic properties of finite fields and polynomial rings to design and analyze the codes, drawing strong connections between classical coding theory and quantum coding theory. The authors investigate bounds on the minimum distance of the constructed codes, a critical parameter determining the error-correcting capability, and explore the use of dual-containing and almost dual-containing codes to improve the performance of quantum codes.

Self-Orthogonal Codes and Improved Distance Bounds

This research presents new constructions of infinite families of quasi-cyclic codes, demonstrating their self-orthogonal properties with respect to both Euclidean and Hermitian inner products. The team determined the dimension of these codes and established a lower bound for their minimum distance, utilising their constituent codes defined over field extensions. Importantly, the lower bound for the minimum distance achieved surpasses the square-root-like bound in certain instances, improving upon previously known results for specific code lengths. The authors acknowledge that determining the parameters of infinite code families is a complex undertaking, and future work will focus on investigating the impact of using maximum distance separable codes as constituent codes, and exploring potential improvements to refine estimates of minimum distances.