Construction of cyclic codes with large minimum distance from power functions over odd characteristic finite fields

2026-06-02

AI Summary

This work addresses the challenge of simultaneously achieving high dimension and large minimum distance for cyclic codes over finite fields of odd characteristic. Building upon power functions with low differential uniformity, the authors generalize Ding et al.'s binary construction to the nonbinary odd-characteristic setting. By leveraging the algebraic structure of cyclic codes together with a careful analysis of their differential properties, they construct several infinite families of \(q\)-ary cyclic codes of length \(q^m - 1\) whose dimensions exceed half the code length and whose minimum distances are provably larger than the square root of the length. Moreover, the exact minimum distances of several such codes are determined for the first time, partially resolving Open Problem 5.31 posed by Ding.

Abstract

Cyclic codes with dimensions exceeding half of the code length and minimum distance greater than the square root of the code length are of significant interest due to their high transmission efficiency and strong error-correcting capability. Such codes are well suited for demanding applications, including communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. Motivated by the work of Ding \cite{P3}, this paper extends the binary framework of Ding and Zhou \cite{P2} to a non-binary setting. By employing power functions with known differential uniformity over finite fields of odd characteristic, we present several infinite families of $q$-ary cyclic codes of length $q^m-1$ with dimensions exceeding $(q^m-1)/2$ and the lower bounds on the minimum distances greater than the square root of the code length, thereby achieving a favorable balance between code rate and error-correcting capability. We also determine the exact minimum distance of some of these codes. Furthermore, we partially resolve Open Problem $5.31$ posed by Ding in \cite{P3}.

Problem

• cyclic codes
• minimum distance
• odd characteristic
• finite fields
• power functions

Innovation

• cyclic codes
• power functions
• odd characteristic finite fields
• minimum distance
• differential uniformity