低差分一致函数循环码
Cyclic codes from low differentially uniform functions
摘要：
循环码由于其高效的编码和解码算法而在消费电子、通信和数据存储系统中具有许多应用。构造循环码的有效方法是序列方法。在他们的文章[Discrete Math.321，2014]和[SIAM J.Discrete Math.27（4），2013]中，Ding和Zhou从有限域上的几乎完全非线性（APN）函数和平面函数构造了几类循环码，并提出了一些关于高度非线性函数的循环码的公开问题。本文通过调查这一研究方向的新见解来关注这些令人兴奋的工作。具体而言，其目标是双重的。第一部分是对前人的一些结果进行了补充，并从APN函数的角度对循环码的一些已知结果给出了正确的证明和说明。第二部分是从一些已知的函数出发，研究了低微分均匀性的循环码。随着这篇文章，我们将提供一些开放的问题在文献中提出的答案。第一个问题涉及开放问题1，由Ding和Zhou在离散数学中提出。321，2014。另外两个是由Ding在[SIAM J. Discrete Math.27（4），2013]中提出的开放问题5.16和5.25。
英文摘要：
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].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
