数论变换的几何表示
Geometrical Representation for Number-theoretic Transforms
摘要：
这篇短文介绍了二进制（或三进制）序列的几何表示。所提出的表示是链接到多变量数据绘制根据雷达图。作为一个说明性的例子，最近提出的二进制汉明变换的几何解释。证明了标准Hamming码$\mathcal{H}(N=7,k=4,d=3)$的码字在Hamming变换下是不变向量。这些不变量是二进制汉明变换的特征向量。这些图像总是刻在一个统一的边的正多边形中，类似于三角形的玫瑰花瓣和/或“荆棘”。还示出了基于在$\operatorname{GF}(3)$上扩展的Golay $\mathcal{G}(N=12, k=6, d=6)$码的三元Golay变换的几何表示。这种方法是作为一种替代表示有限长序列在有限素域。
英文摘要：
This short note introduces a geometric representation for binary (or ternary) sequences. The proposed representation is linked to multivariate data plotting according to the radar chart. As an illustrative example, the binary Hamming transform recently proposed is geometrically interpreted. It is shown that codewords of standard Hamming code $\mathcal{H}(N=7,k=4,d=3)$ are invariant vectors under the Hamming transform. These invariant are eigenvectors of the binary Hamming transform. The images are always inscribed in a regular polygon of unity side, resembling triangular rose petals and/or ``thorns''. A geometric representation of the ternary Golay transform, based on the extended Golay $\mathcal{G}(N=12, k=6, d=6)$ code over $\operatorname{GF}(3)$ is also showed. This approach is offered as an alternative representation of finite-length sequences over finite prime fields.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
