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.

翻译：我们引入了一种汉明型夹角函数 $$\mathrm{angle}_H(u,v):= \min_{c \in \mathbb{F}_q^n} d_H(u, cv)$$，定义于 $\mathbb{F}_q^n$ 中非零向量对上，并证明了该函数在标量乘法下满足所有三条度量公理。函数 $\mathrm{angle}_H$ 在任一参数的非零标量乘法下保持不变，因此可诱导出射影空间 $\mathbb{P}(\mathbb{F}_q^n)$ 上一个真正的整数值度量。作为具体应用，我们证明了线性码唯一译码定理的一个\emph{夹角}（或\emph{射影}）版本：若 $\mathrm{angle}_H(u, C\setminus\{0\}) < d/2$，其中 $d$ 为线性码 $C$ 的最小距离，则 $C$ 中与 $u$ 最近的方向在非零标量乘法意义下是唯一的。随后，我们讨论了这一夹角观点与Reed–Solomon码邻近间隙研究计划之间的关系。据我们所知，这是首次尝试定义有限域上向量的夹角概念，并从几何、编码理论和密码学等多个视角对其进行阐释。