Construction of cyclic codes with large minimum distance from power functions over odd characteristic finite fields
2026-06-02 • Information Theory
Information Theory
AI summaryⓘ
The authors build on previous work by Ding to create new types of error-correcting codes called cyclic codes, but for a wider range of cases beyond just binary codes. These new codes have a good balance: they send lots of data efficiently while still being able to detect and fix many errors. They use special mathematical functions over finite fields with odd characteristics to achieve this. The authors also solve part of an open problem posed by Ding related to these codes.
cyclic codeserror-correcting codesfinite fieldsminimum distancedimensiondifferential uniformityq-ary codescode lengthpost-quantum cryptographyopen problem
Authors
Mrinal Kanti Bose, Abhay Kumar Singh
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}.