有限环上N-重量码及其应用研究

有限环上N(≥2)-重量码与相对N(≥1)-重量码在构造最优码以及秘密共享方案等方面有着重要的应用，因此，近年来是编码与密码学领域中的重要热点之一。本项目将研究有限环上N-重量码和相对N-重量码及其在编码与密码学中的应用。首先，将研究有限环上N-重量线性码的结构，以此构造有限域上达到Plotkin界和Griesmer界的若干类线性码或非线性码；构造有限环上的射影2-重量循环码，并利用有限环上射影2-重量码生成强正则图。其次，利用有限环和有限域上N(≥2)-生成元广义准扭码和2-重量码构造新的线性码或非线性码。最后，利用有限环上的2-重量码确定秘密共享方案的访问结构；构造几类有限环和有限域上新的相对N-重量码，并将其结果应用于第二类窃密信道和秘密共享方案的构造。本课题的研究成果将丰富纠错码理论，为实现信息可靠传输提供重要理论保障，它的深入研究势必推动信息技术乃至国民经济的发展。

Codes with weight N(≥2) and relative weight N(≥1) over finite rings can be used to construct optimal codes and secret sharing schemes. Therefore these codes have been actively pursued in the coding theory and cryptography. In this project, we study N-weight and relative N-weight codes over finite rings, focusing on their applications to coding theories and cryptography as well. Firstly, by studying the structures of N-weight linear codes over finite rings, this project is devoted to constructing some classes of linear codes or nonlinear codes over finite fields, which can arrive at the Plotkin bound and Griesmer bound. The project also constructs projective 2-weight cyclic codes over finite rings and generates strongly regular graphs by using projective 2-weight codes over finite rings. Moreover, this project constructs new linear codes or non-linear codes by applying N(≥2)-generator generalized quasi-twisted codes and 2-weight codes over finite rings and finite fields. Finally，this project determines the access structures of secret sharing schemes by applying 2-weight codes over finite rings, and constructs several types of new relative N-weight codes over finite rings and finite fields which are then used in the wire-tap channel of type II and the.construction of secret sharing schemes. This project enriches error-correcting theory and guarantees reliability of information transmission in theory. Thorough study of topics in this project is of fundamental importance to promote the development of information technology and national economy.