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对于Fq+vFq+v2Fq以上的双循环代码的一些结果
Tenghui Deng1,2,3,4, Jing Yang1
1Department of Mathematics, Tsinghua University, Beijing 100084, China.
Entropy (Basel, Switzerland)
|September 27, 2024
概括
本研究探讨了具有奇特特征的有限场上的双循环代码,重点关注它们的结构和生成器多项式. 研究人员在这些代码和它们的双元代码之间建立了关系,从而实现了最佳的代码构建.
科学领域:
- 编码理论编码理论
- 有限字段 有限字段
- 代数几何几何学的几何学
背景情况:
- 双循环代码是编码理论中重要的代码类,提供独特的代数结构.
- 了解有限非链环上的代码对于开发先进的错误纠正技术至关重要.
- 之前的研究已经探索了各种类型的周期码,但在特定环上的双周期码仍然是研究的一个活跃领域.
研究的目的:
- 在有限环Fq+vFq+v2Fq上研究双循环代码的属性,其中Fq具有奇特特征.
- 为了确定这些代码的生成器多项式和生成器矩阵.
- 建立双循环代码的生成元多项式及其双代码之间的关系.
主要方法:
- 该研究采用代数方法来分析环Fq+vFq+v2Fq的结构,并指出其与Fq×Fq×Fq的同型.
- 生成器多项式和矩阵是使用已建立的编码理论技术衍生出来的.
- 计算双代码的生成多项式,并确定其与原始代码的生成多项式的关系.
主要成果:
- 这篇论文详细分析了指定有限环上的双循环代码.
- 为生成器多项式和矩阵提供了明确的公式.
- 证明了双循环代码和它们的双代码的生成元多项式之间的明显关系.
结论:
- 这些发现有助于更深入地理解对有限非链环的代数代码.
- 由此产生的关系有助于构建和分析这些代码.
- 作为一个应用,该研究成功地在F3上构建了最佳代码,展示了理论结果的实际相关性.
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