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].
翻译:Cyclic代码在消费电子、通信和数据存储系统中有许多应用,原因是其有效编码和解码算法,构建循环代码的有效方法是序列法,其条款[Discrete Math. 321,2014] 和[SIAM J. Discrete Math. 27(4),2013] 、Ding和Zhou分别从一些几乎完美的非线性功能和固定字段平面功能平面功能中构建了几类循环代码,并提出了一些关于高度非线性功能的循环代码的公开问题。这一文章侧重于这些令人振奋的工作,调查了这一研究方向的新见解。具体地说,其目标有两个。第一个是提供一些以往结果的补充,并就一些已知的关于亚光度函数的循环代码提供正确的证据和说明。第二个是从某些已知功能中研究处理低差异统一性的循环代码。除了这一条外,我们将为文献中介绍的一些公开问题提供答案。第一个问题涉及Ding和Zhou在Discre Mathal中提议的开放问题1号,2014年第5号M。