构造有限域上最优线性码和低差分函数一直是编码学和密码学研究的重要内容之一。从有限域上多项式函数构造的线性码，其码的性质往往与多项式函数的差分等密码学性质密切相关。本项目拟综合运用代数学、组合学以及有限域理论来研究有限域上最优线性码和低差分函数的构造问题。具体研究问题如下：(1) 研究在b-距离意义下最优线性码的设计与分析，包括b-距离较大的MDS b-symbol线性码的构造，较少b-重量的线性码的构造，线性码的b-距离达到Hamming距离b倍的充要条件的刻画等。(2) 提出新的基于定义集的线性码构造方法，期望获得具有高码率的最优线性码，并对构造的线性码进行Hamming重量分析。(3) 利用有限域上一些特殊代数曲线来构造低差分函数，并研究它们的差分谱，同时讨论运用这些低差分函数来设计性能优良的线性码。本项目研究成果可为数字信息传输和存储的差错控制提供理论支持，并丰富纠错码理论。

The construction of optimal linear codes and low differential uniformity functions over finite fields is an important research field in coding theory and cryptography. Properties of linear codes constructed from polynomial functions over finite fields are often closely related to the differential uniformity and other cryptographic properties of polynomial functions. In this proposal, we will concentrate our study on constructions of linear codes and low differential uniformity functions over finite fields by using the theory of algebra, combinatorics and the theory of finite fields. The main contents are as follows: (1) We study the design and analysis of optimal linear codes under the b-distance, including the constructions of MDS b-symbol linear codes with large b-distance, the constructions of linear codes with a few b-weight. We also give the characterization of the necessary and sufficient conditions that the b-distance of linear codes are as much as b times of their Hamming distance. (2) A new method of constructing linear codes based on defining sets will be proposed. Our goal is to obtain optimal linear codes with high bit rate; furthermore, the Hamming weight structure of these codes will be analyzed. (3) Some special algebraic curves over finite fields will be used to construct low differential uniformity functions, and their differential spectrums are studied. Moreover, linear codes with good parameters are designed by using these low differential functions. The research results of this proposal can provide theoretical support for the error control of digital information transmission and storage, and also can enrich the theory of error correcting codes.