Cyclic codes from low differentially uniform functions

TL;DR

This paper investigates cyclic codes derived from functions with low differential uniformity, providing new insights, correcting previous results, and addressing open problems related to their construction and properties.

Contribution

It offers new theoretical insights into cyclic codes from low differential uniformity functions, correcting past results and solving open problems in the field.

Findings

Corrected proofs and statements for cyclic codes from APN functions

New results on cyclic codes from low differential uniformity functions

Answers to open problems in the literature

Abstract

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles [Discrete Math. 321, 2014] and [SIAM J. Discrete Math. 27(4), 2013], Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall provide answers to some open problems presented in the literature. The first one concerns Open Problem 1, proposed by Ding and Zhou in Discrete Math. 321, 2014. The two others are Open Problems 5.16 and 5.25, raised by Ding in [SIAM J. Discrete Math. 27(4), 2013].