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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (1): 127-144.doi: 10.3969/j.issn.1005-3085.2024.01.008

• • 上一篇    下一篇

基于二次高斯和的增信码重量分布

衡子灵,  陈辅灵,  李德祥,  刘奋进   

  1. 长安大学理学院，西安 710064
  • 收稿日期:2021-04-04 接受日期:2021-07-23 出版日期:2024-02-15 发布日期:2024-04-15
  • 基金资助:
    国家自然科学基金 (11901049)；陕西省自然科学基础研究计划 (2020JQ-343; 2021JM-149)；陕西省高校科协青年人才托举计划 (20200505).

Weight Distribution of Augmented Codes Based on Quadratic Gauss Sums

HENG Ziling,  CHEN Fuling,  LI Dexiang,  LIU Fenjin   

  1. School of Science, Chang'an University, Xi'an 710064
  • Received:2021-04-04 Accepted:2021-07-23 Online:2024-02-15 Published:2024-04-15
  • Supported by:
    The National Natural Science Foundation of China (11901049); the Natural Science Basic Research Program of Shaanxi Province (2020JQ-343; 2021JM-149); the Young Talent Fund of University Association for Science and Technology in Shaanxi Province (20200505).

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摘要：

循环码是一类具有高效编码和译码算法的重要线性码，在通信和数据存储系统等领域被广泛应用。重量分布是循环码的重要研究对象，如何构造具有较少重量的循环码是编码理论中的重要研究问题。借助于二次高斯和、有限域上特征标的性质等数论工具研究了一类循环码的增信码，给出了增信码的重量分布和参数。结果表明，这类增信码是射影循环码且具有较少的重量。该增信码与原码相比有更高的传输效率且其对偶码关于球填充界最优或几乎最优。此外，还给出了原码的完全重量分布。

关键词: 循环码, 增信码, 高斯和, 重量分布, 完全重量分布

Abstract:

Cyclic codes are an important subclass of linear codes as they have efficient encoding and decoding algorithms and they are widely used in many areas including communication and data storage. The weight distribution is an interesting research subject of cyclic codes. How to construct cyclic codes with only a few weights is an important research problem in the coding theory. To solve this problem, the augmented codes of a class of cyclic codes are investigated based on quadratic Gauss sums and some basic properties of characters over finite fields. The weight distribution and parameters of the augmented code are presented. It turns out that the augmented codes are projective cyclic codes with only a few weights. Compared with the original cyclic code, the augmented codes have higher transmission efficiency. The duals of the augmented codes are optimal or nearly optimal according to the sphere-packing bound. As a byproduct, the complete weight distribution of the original cyclic codes is also given.

Key words: cyclic code, augmented code, Gauss sum, weight distribution, complete weight distribution

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