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Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes

Fernando Hernando, Gregorio Quintana-Ortí

Abstract

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.

Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes

Abstract

We present a new general method for performing basic arithmetic in the finite field~ for any prime by using traditional binary operations over~. 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 and . 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.

Paper Structure

This paper contains 28 sections, 14 equations, 7 figures, 5 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Additions in $\mathbb{F}_p$ via $\mathbb{F}_2$
  3. Arithmetic for the case $p=2^r-1$
  4. Arithmetic over $\mathbb{F}_3$
  5. Arithmetic over $\mathbb{F}_7$
  6. Arithmetic for the case $p\ne 2^r-1$
  7. Minimum Hamming distance of a random linear code
  8. Brouwer-Zimmermann algorithm
  9. Algorithms and implementations
  10. Data storage
  11. Hamming weight of linear combinations in $\mathbb{F}_p$
  12. Optimization of products: circular shift optimization.
  13. Reducing addition cost via isometry
  14. Case $\mathbb{F}_3$:
  15. Case $\mathbb{F}_p$ with $p = 2^r - 1$:
  16. ...and 13 more sections

Figures (7)

  • Figure 1: Speedups of our new implementations with respect to Magma for all matrices in the dataset 3 of $\mathbb{F}_7$. Each subplot contains the results of one subdataset.
  • Figure 2: Times in seconds (left) and speedups (right) for a random sample of matrices in subdataset 3_a of $\mathbb{F}_7$. In these plots Magma time is $[1, 10)$. The horizontal axis shows the matrices assessed with their dimensions ($k \times n$) and their distance (d).
  • Figure 3: Times in seconds (left) and speedups (right) for a random sample of matrices in subdataset 3_b of $\mathbb{F}_7$. In these plots Magma time is $[10, 100)$. The horizontal axis shows the matrices assessed with their dimensions ($k \times n$) and their distance (d).
  • Figure 4: Times in seconds (left) and speedups (right) for a random sample of matrices in subdataset 3_c of $\mathbb{F}_7$. In these plots Magma time is $[100, 1000)$. The horizontal axis shows the matrices assessed with their dimensions ($k \times n$) and their distance (d).
  • Figure 5: Times in seconds (left) and speedups (right) for a random sample of matrices in subdataset 3_d of $\mathbb{F}_7$. In these plots Magma time is $[1000, 10000)$. The horizontal axis shows the matrices assessed with their dimensions ($k \times n$) and their distance (d).
  • ...and 2 more figures

Theorems & Definitions (3)

  • Example 1
  • Example 2
  • Example 3