线性码的正交包及相关问题研究

线性码的正交包在密码学等领域具有很多重要的应用，近年来备受人们关注。在编码理论中，人们希望构造具有良好参数的码，因此基于正交包的线性码研究是一个非常有意义的研究课题。本项目主要利用代数、数论等数学工具研究线性码（特别是循环码、常循环码、BCH码等）正交包的刻画与构造，分析相关码的参数和正交包的性质，并应用于设计掩码方案和构造纠缠辅助量子纠错码。具体如下：1）刻画具有固定维数正交包的线性码，得到最优或几乎最优码；2）研究线性码的正交包，给出较大维数的正交包或自正交码；3）利用线性码对设计掩码方案并确定相应的安全参数；4）构造最优或极大纠缠的纠缠辅助量子纠错码。本项目研究成果将进一步发展和完善纠错码理论，并对编码理论在密码学等领域的应用研究产生积极的促进作用。

The hulls of linear codes have many important applications in cryptography, and have attracted much attention in recent years. Since codes with good parameters are desirable to be constructed in coding theory, the research of linear codes based on their hulls has been an interesting topic. The objective of this project is to employ algebra and number theory to investigate the characterizations and constructions of the hulls of linear codes (especially cyclic codes, constacyclic codes, and BCH codes etc.), analyse the parameters of the codes and the properties of the hulls, and use the codes to design masking schemes and construct entanglement-assisted quantum error-correcting codes. The details are described as follows: 1)characterizing the linear codes via their hulls with fixed dimensions, and obtaining optimal or almost optimal codes; 2) investigating the hulls of linear codes and presenting the hulls or self-orthogonal codes with large dimensions; 3) employing the pairs of linear codes to design masking schemes and determining the security parameters; 4) constructing entanglement-assisted quantum error-correcting codes with optimal parameters or maximal entanglement. The project will not only contribute to the theory of error-correcting codes, but also promote the applications of coding theory in cryptography.