有限环上广义准多项式循环码的理论及其应用研究

有限环(域)上广义准多项式循环码在构造新参数的纠错码以及在信息安全中有着重要的应用，因此是编码与密码中的重要研究问题。本项目将围绕广义准多项式循环码的理论及其应用展开研究。首先，研究有限环(域)上广义准多项式循环码的结构理论，定义环上保距的Gray映射并构造达到Plotkin界和Griesmer界的几类新的纠错码。其次，研究满足Gilbert-Varshamov界的广义准多项式循环码的充要条件，构造有限环(域)上几类渐进好的广义准多项式循环码。最后，研究广义准多项式循环码的结构性质，搜索具有新参数的FSD码、Isodual码和LCD码，并将FSD码与Isodual码分别应用于区组设计与Isodual格的构造，尝试将LCD码应用于抵抗侧信道攻击和故障注入攻击的防御实现。本项目的研究将丰富纠错码理论，为实现信息可靠传输提供重要理论保障，它的深入研究必将推动信息技术乃至国民经济的发展。

Generalized quasi-polynomial cyclic codes over finite rings (fields) have important applications in constructing error-correcting codes with new parameters over finite fields and in information security, so they are important research problems in coding and cryptography. This project will focus on the theory and application of generalized quasi-polynomial cyclic codes. Firstly, we study the structure theory of generalized quasi-polynomial cyclic codes over finite rings and finite fields, define the distance preserving Gray mapping over rings, and construct several families of new error-correcting codes that reach the Plotkin and Griesmer bounds. Secondly, we study the necessary and sufficient conditions of generalized quasi-polynomial cyclic codes satisfying the Gilbert-varshamov bound, and construct several classes of asymptotically good generalized quasi polynomial cyclic codes over finite rings (fields). Finally, we study the structure properties of generalized quasi-polynomial cyclic codes to search for LCD codes, Isodual codes and FSD codes with new parameters, and apply the FSD codes and Isodual codes to the construction of block designs and Isodual lattices, respectively. We also try to apply the isodual codes to the defense implementation against side-channel attacks and fault injection attacks. The research of this project will enrich the theory of error-correcting codes and provide important theoretical guarantee for the reliable transmission of information. Its in-depth research will promote the development of information technology and even the national economy.