论文标题
来自低差异均匀功能的环状代码
Cyclic codes from low differentially uniform functions
论文作者
论文摘要
由于其有效的编码和解码算法，环保代码在消费电子，通信和数据存储系统中具有许多应用。序列方法是一种有效的构建环形代码的方法。在他们的文章中[离散数学。 321，2014]和[Siam J.离散数学。 27（4），2013年]，丁和周从有限领域的几乎完美的非线性（APN）函数和平面函数中构建了几类环状代码，并在高度非线性函数上对周期性代码提出了一些开放问题。本文通过调查了这个研究方向的新见解，重点介绍了这些令人兴奋的作品。具体而言，其目标是双重的。首先是提供一些以前的结果的补充，并在APN函数上对循环代码上的某些已知元素提供正确的证明和陈述。第二个是从某些已知功能处理低差异均匀性的循环代码。与本文一起，我们将为文献中提出的一些开放问题提供答案。第一个涉及开放问题1，该问题1是在离散数学中提出的。 321，2014。另外两个是空旷的问题5.16和5.25，在[Siam J. Increst Math In In ding中提出。 27（4），2013年]。
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].
