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Cyclic codes have many applications in consumer electronics, communicationand data storage systems due to their efficient encoding and decodingalgorithms. An efficient approach to constructing cyclic codes is the sequenceapproach. In their articles [Discrete Math. 321, 2014] and [SIAM J. DiscreteMath. 27(4), 2013], Ding and Zhou constructed several classes of cyclic codesfrom almost perfect nonlinear (APN) functions and planar functions over finitefields and presented some open problems on cyclic codes from highly nonlinearfunctions. This article focuses on these exciting works by investigating newinsights in this research direction. Specifically, its objective is twofold.The first is to provide a complement with some former results and presentcorrect proofs and statements on some known ones on the cyclic codes from theAPN functions. The second is studying the cyclic codes from some knownfunctions processing low differential uniformity. Along with this article, weshall provide answers to some open problems presented in the literature. Thefirst 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].

Cyclic codes from low differentially uniform functions