在编码与序列设计中有很多代数与组合的方法。其中代数的方法主要有：（有限）群、（有限）环、（有限）域等代数结构的理论，以及 Galois理论，特征和等。如在群码（阵列码)、循环码的研究中，人们关注某些代数结构的直和分解和极小幂等元，以及相应的计数问题等。组合的方法主要有：区组设计，包括差集，差族，以及有限几何，图论等。在这些方法中，特征和方法占有中心的地位。本课题主要利用一些特征和方法，研究：(1)利用特征和方法，设计偶特征域上新的密码函数，构造新的优线性码；(2)利用分圆，整二次型，模形式等代数、几何、以及数论的工具构造优的码本和优的跳频序列、优的存取结构等, 并利用特征和确定这些结构的参数。这些工作将为保密通信等领域提供理论支持，同时在量子信息处理、压缩感知等领域也有重要的应用。

There are many algebraic and combinatorial methods in coding and sequences designing. Algebraic methods include the theory of (finite) groups, (finite) rings, (finite) fields, and the theory of Galois, character sums etc. For example, in the researching on group codes (array codes), cyclic codes, people concern the decomposition of certian algebraic structure and minimum idempotents, and the enumeration of them. Combinatorial methods include block designs, difference sets, difference families, and finite geometry, graph theory etc. Among these methods, the character sum method stands in the center stage. The main purpose of this project is to use character sum method to study (1) how to design new cryptographic functions and construct new optimal linear codes; (2)ultilize the tools of algebra, geometry, and number theory such as cyclotomy, quadratic forms and modulo forms to construct new optimal codebooks and optimal frequency-hopping sequences, and access structures, and determine the parameters of those structures. Our work will provide theoretic support to security communications, and have important applications in other areas such as quantum information processing, compressed sensing etc.