有限环上N-重量（循环）迹码的研究及其应用

本项目利用有限域，数论，组合数学和环上编码理论等多种方法，并在有限环上N-重.量迹码方面作出的成果上，考虑研究有限环上N-重量（循环）迹码的存在性和唯一性.问题；讨论其Gray像的线性性，期望得到几类有限域上性能良好的纠错码。其次，研.究有限环上N-重量（循环）迹码的结构，给出其重量分布（或重量计数器）和对应的.搜索算法，研究N-重量线性码的构造方法（比如密码学中的boolean函数、二次bent函.数等如何构造非主理想环上的N-重量（循环）迹码），进一步研究其Gray像的线性性，.期望得到性能良好的纠错码。最后，研究上述有限环上的N-重量（循环）迹码.应用于认证码、图论、几何和密钥方案的设计等。

In this paper, the existence and uniqueness of N-weight (cyclic) traces on finite rings are studied based on the results of N-weight (cyclic) traces on finite rings by using various methods such as finite field, number theory, combinatorial mathematics and coding theory on rings. The linearity of Gray image is discussed and several kinds of error-correcting codes with good performance in finite fields are expected to be obtained. Second, the structure of N-weight (cyclic) trace codes on finite rings is studied, and its weight distribution counter (or weight) and the corresponding search algorithm are given. What's more, the paper study the structure of the linear code N - weight method (such as the Boolean function in cryptography, secondary bent function, and how to construct N -weight (cyclic) trace codes), further study on its Gray, like linear, hoping for a good performance of error correction codes. Finally, the application of N-weight (cyclic) trace codes on the above finite ring to authentication codes, graph theory, geometry and key scheme design is studied.