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Computing efficiently a parity-check matrix for Zps-additive codes

Cristina Fernández-Córdoba, Adrián Torres, Carlos Vela, Mercè Villanueva

TL;DR

This work addresses the challenge of efficiently computing parity-check matrices for $\mathbb{Z}_{p^s}$-additive codes from generator matrices in standard form. It introduces two constructive methods: a block-minor approach based on the reduced associated matrix $G^{RA}$ and a recurrence-driven iterative method that reuses intermediate results. The authors provide detailed algorithmic descriptions, implement them in Magma, and compare their performance to Magma's general parity-check function, showing substantial speedups, especially for the iterative method. The work also offers explicit time-complexity analyses and proposes a MagmaZps package to facilitate practical usage and future extensions to mixed alphabets and related code families.

Abstract

The Zps-additive codes of length n are subgroups of Zps^n , and can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s . In this paper, we show two methods for computing a parity-check matrix of a Zps-additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general. A time complexity analysis is also shown.

Computing efficiently a parity-check matrix for Zps-additive codes

TL;DR

This work addresses the challenge of efficiently computing parity-check matrices for -additive codes from generator matrices in standard form. It introduces two constructive methods: a block-minor approach based on the reduced associated matrix and a recurrence-driven iterative method that reuses intermediate results. The authors provide detailed algorithmic descriptions, implement them in Magma, and compare their performance to Magma's general parity-check function, showing substantial speedups, especially for the iterative method. The work also offers explicit time-complexity analyses and proposes a MagmaZps package to facilitate practical usage and future extensions to mixed alphabets and related code families.

Abstract

The Zps-additive codes of length n are subgroups of Zps^n , and can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s . In this paper, we show two methods for computing a parity-check matrix of a Zps-additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general. A time complexity analysis is also shown.
Paper Structure (8 sections, 10 theorems, 60 equations, 1 figure, 2 algorithms)

This paper contains 8 sections, 10 theorems, 60 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Computation of a parity-check matrix
  3. Performance comparison
  4. Algorithms description
  5. Performance comparison
  6. Computation time analysis
  7. Time complexity analysis
  8. Conclusion

Key Result

Proposition 2.1

Let $A$ be a matrix as in (eq:MatrixA). Then, the determinant of $A$ is given by where $\hat{{\cal S}}_n$ and $J_{\sigma}$ are as in Definition defi:J.

Figures (1)

  • Figure :

Theorems & Definitions (19)

  • Definition 2.1
  • Proposition 2.1
  • Example 2.1
  • Definition 2.2
  • Proposition 2.2
  • Corollary 2.1
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Lemma 2.1
  • ...and 9 more