Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes
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| Format: | Preprint |
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2026
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| _version_ | 1866908926872649728 |
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| author | Hernando, Fernando Quintana-Ortí, Gregorio |
| author_facet | Hernando, Fernando Quintana-Ortí, Gregorio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29942 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes Hernando, Fernando Quintana-Ortí, Gregorio Information Theory 68Q30, 68Q25, 65F99, 11Y16 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. |
| title | Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes |
| topic | Information Theory 68Q30, 68Q25, 65F99, 11Y16 |
| url | https://arxiv.org/abs/2603.29942 |