关于扩展场的矩阵表示 (GF) 和其在矢量线性网络编码中的应用
Hanqi Tang1, Heping Liu1, Sheng Jin1
1School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China.
Entropy (Basel, Switzerland)
|October 25, 2024
概括
本研究概括了有限场的矩阵表示法GF(pL),以增强矢量线性网络编码 (LNC). 它揭示了固有的矩阵相关性,使得查找表更小,并展示了在LNC中对二进制矩阵表示的优势.
科学领域:
- 编码理论编码理论
- 有限场数学的算术
- 网络编码 网络编码
背景情况:
- 有限字段的标准矩阵表示法 GF(pL) / GF(p) 允许算术运算.
- 现有的编码方案在GF(2L) 上的现有实现使用矩阵表示,但忽略了代表不同字段元素的矩阵之间的相关性.
研究的目的:
- 将经典矩阵表示结果从GF{2L}到GF{pL}概括.
- 揭示并利用表格之间的固有相关性,这些表格代表了GF (pL) 元素的不同权力.
- 提高矢量线性网络编码 (LNC) 方案的效率.
主要方法:
- 对GF (pL) 与GF (p) 的矩阵表示的概括.
- 分析表格之间的内在相关性,这些表格代表了场元素的权力.
- 基于矩阵相关性的预先存储的查找表的开发.
主要成果:
- 建立了一个对GF (pL) 乘以GF (p) 的概括矩阵表示.
- 确定并澄清了代表GF (pL) 元素不同权力的矩阵之间的内在相关性.
- 为矩阵表示设计了一个更紧的查找表,从而减少了存储需求.
- 演示二进制矩阵表示在矢量LNC中的优点的理论结果得出.
结论:
- 一般化的矩阵表示和识别的相关性为基于矩阵的有限场算法提供了更透明的理解.
- 拟议的查找表提高了矢量LNC实现的效率.
- 这些发现强调了二进制矩阵表示在高级编码方案中的好处.
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