We present a new general method for performing basic arithmetic in the finite field~$\mathbb{F}_p$ for any prime $p>2$ by using traditional binary operations over~$\mathbb{F}_2$. Our new approach is efficient and competitive with current state-of-art methods. We apply our new arithmetic method to the computation of the minimum Hamming distance of random linear codes for the fields $\mathbb{F}_3$ and $\mathbb{F}_7$. Our new arithmetic method allows to apply new techniques such as the isometric addition that accelerate the computation of the Hamming distance. We have developed implementations in the C programming language for computing the Hamming distance that clearly outperform both state-of-art licensed software and open-source software such as \textsc{Magma} and \textsc{GAP}/\textsc{Guava} on single-core processors, multicore processors, and shared-memory multiprocessors.

翻译：我们提出一种利用$\mathbb{F}_2$上的传统二进制运算，对任意素数$p>2$执行有限域$\mathbb{F}_p$中基本算术运算的新通用方法。该方法在效率上具有竞争力，可与当前最先进的方法相媲美。我们将这一新算术方法应用于$\mathbb{F}_3$和$\mathbb{F}_7$域上随机线性码最小汉明距离的计算中。该算术方法支持引入等距加法等新技术，从而加速汉明距离的计算。我们使用C编程语言实现了汉明距离的计算程序，在单核处理器、多核处理器及共享内存多处理器上，其性能均显著优于当前最先进的商业软件及开源软件（如\textsc{Magma}和\textsc{GAP}/\textsc{Guava}）。