有限域上最优LCD码的构造研究

Linear codes with complementary duals (LCD codes) have many important applications in information theory and cryptography, and have attracted much attention for the past few decades. Since codes with good parameters are desirable to be employed in communication systems, the construction of optimal LCD codes has been an interesting topic in coding theory. The objective of this project is to employ algebra and number theory (group character, exponential sum etc.) to investigate the constructions of LCD codes over finite fields and their parameters, and obtain optimal or nearly optimal LCD codes. The details are described as follows: 1) investigating the properties of LCD codes to derive and improve some theoretical bounds; 2) using group characters, 2-designs, and defining sets to construct LCD codes, and constructing (Hermitian) LCD codes from cyclic codes and constacyclic codes; 3) employing check matrices, cyclotomic cosets, and exponential sums to analyze the parameters of LCD codes, and presenting optimal or nearly optimal LCD codes with respect to the theoretical bounds; 4) considering some applications in cryptography, such as side-channel attacks or fault injection attacks. The project will not only contribute to the theory of error-correcting codes, but also promote the applications of LCD codes in cryptography and communication systems.

LCD码在信息论和密码学中有很多重要的应用，多年来备受人们关注。在通信系统中，人们希望使用具有良好参数的码，因此最优LCD码的构造是编码理论中一个非常有意义的研究课题。本项目主要利用代数、数论等数学工具（主要群特征、指数和等）研究有限域上LCD码的构造和参数，得到最优或几乎最优的LCD码。具体如下：1）研究LCD码的性质，给出或改进码的相关参数的理论界；2）通过群特征、2-设计、定义集构造LCD码，并利用循环码和常循环码构造LCD码（或Hermitian LCD码）；3）利用校验矩阵、分圆陪集、指数和等工具分析LCD码的参数，得到最优或几乎最优的LCD码；4) 研究LCD码在边信道攻击、错误注入攻击等密码学领域中的应用。本项目研究成果将进一步发展和完善纠错码理论，并有望促进LCD码在密码和通信系统中的应用。

本项目主要利用代数、数论等数学理论对LCD码和相关码的构造与参数进行深入研究，给出最优或几乎最优码。本项目取得的主要结果如下：研究了有限域上两类（欧几里得）LCD BCH码，通过分析分圆陪集的性质给出一些码的维数和最小距离的下界；对循环Hermitian LCD码做了系统的研究，给出了两种类型长度下的循环Hermitian LCD码的维数；确定了两类本原BCH码以及它们的扩展码的重量分布，解决了由丁存生教授等在2017年提出的两个问题；基于LCD码（正交包维数为0）给出推广工作，借助本原BCH码和射影BCH码设计距离的上下界给出了两类本原和射影BCH码具有固定维数正交包的若干充要条件，给出几类自正交码和它们的参数。这些研究成果进一步发展和完善了纠错码理论，并对LCD 码在密码学和通信等领域的应用研究产生积极的促进作用。