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Permutation decoding of algebraic geometry codes from Hermitian and norm-trace curves

Monica Lichtenwalner, Hiram H. López, Gretchen L. Matthews, Padmapani Seneviratne

Abstract

Permutation decoding is a process that utilizes the permutation automorphism group of a linear code to correct errors in received words. Given a received word, a set of automorphisms, called a PD set, moves errors out of the information positions so that the original message can be determined. In this paper, we investigate permutation decoding for certain families of algebraic geometry codes. Automorphisms of the underlying curve are used to specify permutation automorphisms of the code. Specifically, we describe permutation decoding sets that correct specific burst errors for one-point codes on Hermitian and norm-trace curves.

Permutation decoding of algebraic geometry codes from Hermitian and norm-trace curves

Abstract

Permutation decoding is a process that utilizes the permutation automorphism group of a linear code to correct errors in received words. Given a received word, a set of automorphisms, called a PD set, moves errors out of the information positions so that the original message can be determined. In this paper, we investigate permutation decoding for certain families of algebraic geometry codes. Automorphisms of the underlying curve are used to specify permutation automorphisms of the code. Specifically, we describe permutation decoding sets that correct specific burst errors for one-point codes on Hermitian and norm-trace curves.
Paper Structure (14 sections, 11 theorems, 81 equations, 1 table)

This paper contains 14 sections, 11 theorems, 81 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutation decoding
  4. Algebraic geometry codes
  5. Code automorphisms from curve automorphisms
  6. Hermitian codes
  7. Automorphisms
  8. Systematic Representation
  9. Permutation Decoding
  10. Norm-trace codes
  11. Automorphisms
  12. Systematic Representation
  13. Permutation Decoding
  14. Conclusion

Key Result

Lemma 2.2

Handbook_chapter Consider an $[n,k,d]$ code $C$ over $\mathbb{F}_q$ with information set $I=[k]$ and parity check matrix $H=[-A^T \mid I_{n-k}] \in \mathbb{F}_q^{n-k \times n}$. For any received word $y \in \mathbb{F}_q^n$, $y\mid_I$ is correct (meaning there are no errors in the first $k$ coordinat

Theorems & Definitions (26)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Proposition 3.2
  • Example 3.3
  • Example 3.4
  • ...and 16 more