基于一般构造的线性码及其应用的研究

Linear codes enjoy a prominent place in the theory of error-correcting codes, which are not only closely related to a number of mathematical areas such as algebra, algebraic function fields, algebraic geometry, algebraic number theory, association schemes, combinatorial designs, finite geometries, finite fields, and number theory, but also have important applications in many areas of engineering and science such as communications systems, computer systems, consumer electronics, quantum physics, and cryptography. .Based on the general constructions, this project is to explore issues of linear codes over finite fields and their applications in cryptography, communications systems and other fields. The objectives of this research are listed in the following: 1) to put forward a new general construction for all linear codes over finite fields, especially for special families of linear codes such as cyclic codes and linear code with complementary dual (LCD codes); 2) to obtain some special families of linear codes with good error correcting capability and a few weights, and determine their parameters (e.g., code length, dimension, minimum distance, weight distribution, complete weight distribution and so on ) by using exponential sums, algebraic, geometric, number-theoretic and other mathematical tools; 3) to consider their applications in the constructions of the optimal (or almost optimal) codebooks, secret sharing schemes and authentication codes.

线性码作为一类重要的纠错码，它不仅与数学理论知识（例如代数、代数函数域、代数几何、代数数论、结合方案、组合设计、有限几何、有限域、数论等）紧密相关，而且在工程和科学的很多领域（例如量子物理、密码学、通信和计算系统等）具有重要的应用。.本课题着重研究有限域上基于一般构造的线性码及其在密码学和通信等领域中的应用问题，具体包括：1）针对有限域上所有的线性码给出新的一般构造方法，尤其是对具有某些特殊性质的线性码（例如循环码、LCD码）；2）构造出新的带有很好纠错能力和很少非零重量的最优（或几乎最优）线性码，并使用指数和、代数、几何、数论等数学知识去确定它们的参数（例如：码长、维数、最小距离、重量分布、完全重量分布等）；3）考虑线性码在构造最优（或几乎最优）码本、密钥共享方案和认证码等通信和密码学领域中的应用。

本项目主要研究线性码的构造及其应用。关于最优（或几乎最优）线性码的构造，Ding和Niederreiter给出有限域上线性码的一般构造，通过分析他们的构造方法，我们构造了两类带有很少非零重量的线性码，并且确定了它们的参数，其中一些码是最优或几乎最优的。关于线性码的应用问题，Ding等人基于线性码构造了一些最优或几乎最优码本、密钥共享方案、认证码，我们对此应用问题进一步研究，并构造了一些几乎最优码本，所得结论推广了衡子灵的相关结果。此外，我们探讨了线性码在组合设计方面的应用，通过一些特殊线性码和函数构造了一些2设计，并确定了它们的参数。.上述主要结果发表在Designs,Codes and Cryptography、Applicable Algebra in Engineering, Communication and Computing、Cryptography and Communications、IEEE Communications Letters等期刊上。还有一篇论文已投到Advances in Mathematics of Communications上，目前正在评审。

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