Angle Between Two Vectors over Finite Fields and an Application to Projective Unique Decoding

TL;DR

This paper introduces a new angular metric for vectors over finite fields, applies it to unique decoding of linear codes, and explores its connections to geometry, coding theory, and cryptography.

Contribution

It defines the first angle concept for finite field vectors and demonstrates its use in projective decoding and related areas.

Findings

Established an angular metric satisfying metric axioms up to scalar multiplication.

Proved a unique decoding theorem based on the angular metric.

Connected the angular viewpoint to Reed--Solomon codes and cryptography.

Abstract

We introduce a Hamming-type angular function $$\mathrm{angle}_H(u,v):= \min_{c \in \mathbb{F}_q^n} d_H(u, cv)$$ on pairs of nonzero vectors in $\mathbb{F}_q^n$ and show that it satisfies all three metric axioms up to scalar multiplication. The function $\mathrm{angle}_H$ is invariant under nonzero scalar multiplication in either argument and therefore descends to a genuine integer-valued metric on the projective space $\mathbb{P}(\mathbb{F}_q^n)$. As a concrete application, we prove an \emph{angular} (or \emph{projective}) version of the unique-decoding theorem for linear codes: if $\mathrm{angle}_H(u, C\setminus\{0\}) < d/2$, where $d$ is the minimum distance of the linear code $C$, then the closest direction in $C$ to $u$ is unique up to nonzero scalar multiplication. We then discuss how this angular viewpoint relates to the proximity-gap programme for Reed--Solomon codes. To the best of our knowledge, this is the first attempt to define an angle notion for vectors over finite fields and interpret it from several perspectives, including geometry, coding theory, and cryptography.