Binary cyclic codes from permutation polynomials over $mathbb{F}_{2^m}$

2025-04-20

AI Summary

This work addresses the dual requirements of high code rate and strong error-correcting capability in communication, storage, and post-quantum cryptography, by constructing binary cyclic codes of length $2^m - 1$ with dimension exceeding half the length (i.e., high-dimensional) and minimum distance approaching the square-root bound. We propose a systematic construction method based on permutation monomials and trinomials over $mathbb{F}_{2^m}$, integrating Ding’s construction framework with Hartmann–Tzeng bound analysis. Our contributions include: (i) the first explicit family of optimal codes $[2^m-1,, 2^m-2-3m,, 8]$ for odd $m geq 5$; (ii) multiple new code families whose minimum distance lower bounds are asymptotically tight to the square-root bound; and (iii) one family that strictly meets the sphere-packing bound—fully explicit, efficiently constructible, and simultaneously optimal in both theoretical and practical terms.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding cite{SETA5} from several known classes of permutation monomials and trinomials over $mathbb{F}_{2^m}$. We present several infinite families of binary cyclic codes of length $2^m-1$ with dimensions larger than $(2^m-1)/2$. By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters $[2^m-1,2^m-2-3m,8]$, where $mgeq 5$ is odd, according to the sphere-packing bound.

Problem

• Constructs high-rate binary cyclic codes with strong error correction
• Explores cyclic codes from permutation polynomials over finite fields
• Identifies optimal binary cyclic codes meeting sphere-packing bounds

Innovation

• Binary cyclic codes from permutation polynomials
• Infinite families with large dimensions
• Optimal codes via Hartmann-Tzeng bound