Geometrical Representation for Number-theoretic Transforms

TL;DR

This paper introduces a geometric visualization method for binary and ternary sequences, linking them to multivariate data plotting, and demonstrates its application to Hamming and Golay transforms, revealing invariant codewords as geometric patterns.

Contribution

It presents a novel geometric representation for finite sequences over prime fields, connecting sequence transforms to visual patterns and invariance properties.

Findings

Codewords of Hamming code are invariant vectors under the Hamming transform.

Invariant vectors correspond to eigenvectors of the transform.

Sequences are represented as inscribed polygons resembling rose petals.

Abstract

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.