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

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.