伽罗华环上指数和及其在编码理论中的应用

项目名称： 伽罗华环上指数和及其在编码理论中的应用

项目编号： No.11501156

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 李锦

作者单位： 合肥工业大学

项目金额： 18万元

中文摘要： 有限域上指数和在编码理论中已有广泛的应用，有限环上指数和作为有限域上指数和的推广，也开始成为研究纠错码的重要工具。然而，有限环上指数和的研究还未充分展开。本项目将研究伽罗华环上指数和，主要考虑伽罗华环GR(p^s,r)(s≥2)上高斯和、雅可比和等。在申请人博士毕业论文中，已考虑了伽罗华环GR(p^2,r)上高斯和与雅可比和，证明了对于所有非平凡的情况，伽罗华环GR(p^2,r)上的高斯和与雅可比和都可以化简到有限域F_{p^r}上。本项目通过进一步改进研究方法，建立伽罗华环GR(p^s,r)(s≥2)上指数和与有限域上指数和的联系。在此基础上，利用有限域及伽罗华环上指数和理论，研究环Z_p^s(s≥2)上几类线性码的重量分布。本项目不仅进一步充实和完善了有限环上指数和理论，而且给出了伽罗华环上指数和在编码理论中的一些应用，对研究一般有限交换环上指数和及其应用起着积极作用。

中文关键词： 指数和；伽罗华环；纠错码；重量分布；有限域

英文摘要： Since exponential sums over finite fields have many applications in coding theory, exponential sums over finite rings as generalize of exponential sums over finite fields, exponential sums over finite rings have become very important tools to study error-correcting codes. However, we know very little about the exponential sums over finite rings...The present project aims to explore the exponential sums over Galois rings GR(p^s,r)(s≥2), and primarily research Gauss sums and Jacobi sums and so on. In the applicant’s PH.D. thesis，the Gauss sums and Jacobi sums over Galois ring GR(p^2,r), show that the values of these sums can be reduced to the Gauss sums and Jacobi sums over finite field F_{p^r} for all non-trivial cases. This project will improve research methods, establish the relation between exponential sums over Galois rings and the exponential sums over finite fields, show that the values of exponential sums over Galois rings can be reduced to the exponential sums over finite fields. Then we determine the weight distributions of some linear codes over ring Z_p^s(s≥2) by exponential sums over Galois ring...The project enriches the theory of exponential sums over Galois ring, and give their applications in coding theory. The project will play a positive role in research the exponential sums over finite communicative ring and their applications.

英文关键词： Exponential sums;Galois ring;error correcting codes;weight distribution;finite field