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The permutation automorphism groups of irreducible cyclic codes

Tao Feng, Henk D. L. Hollmann, Weicong Li, Qing Xiang

Abstract

The study of permutation automorphism groups of cyclic codes is a central topic in algebraic coding theory. A cyclic code over $\mathbb{F}_q$ is called irreducible if its check polynomial is irreducible over $\mathbb{F}_q$. Such a code is standard if its permutation automorphism group is equal to the group generated by the cyclic shift and the Frobenius automorphism, and non-standard otherwise. In this paper, we give a complete classification of all non-standard non-degenerate irreducible cyclic codes, using the classification of finite simple groups. Our result shows that, apart from a small number of explicit exceptional families and their descendants under certain secondary constructions, every non-degenerate irreducible cyclic code is standard, and up to four explicit exceptions, every degenerate cyclic code is non-standard. This classification has several consequences. First, it yields a general description of non-standard linear recurring sequence subgroups, extending the earlier work of Brison and Nogueira; secondly it establishes the Schmidt-White conjecture for all non-standard irreducible cyclic codes. Moreover, our results provide strong evidence in support of the conjecture of Berger and Charpin that almost all cyclic codes are standard.

The permutation automorphism groups of irreducible cyclic codes

Abstract

The study of permutation automorphism groups of cyclic codes is a central topic in algebraic coding theory. A cyclic code over is called irreducible if its check polynomial is irreducible over . Such a code is standard if its permutation automorphism group is equal to the group generated by the cyclic shift and the Frobenius automorphism, and non-standard otherwise. In this paper, we give a complete classification of all non-standard non-degenerate irreducible cyclic codes, using the classification of finite simple groups. Our result shows that, apart from a small number of explicit exceptional families and their descendants under certain secondary constructions, every non-degenerate irreducible cyclic code is standard, and up to four explicit exceptions, every degenerate cyclic code is non-standard. This classification has several consequences. First, it yields a general description of non-standard linear recurring sequence subgroups, extending the earlier work of Brison and Nogueira; secondly it establishes the Schmidt-White conjecture for all non-standard irreducible cyclic codes. Moreover, our results provide strong evidence in support of the conjecture of Berger and Charpin that almost all cyclic codes are standard.
Paper Structure (15 sections, 27 theorems, 36 equations, 2 tables)

This paper contains 15 sections, 27 theorems, 36 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Group theoretic notation and results
  5. Standard and non-standard non-degenerate cyclic codes
  6. Degenerate cyclic codes
  7. Non-degenerate irreducible cyclic codes
  8. Examples of non-standard non-degenerate irreducible cyclic codes
  9. A new family of non-standard non-degenerate irreducible cyclic codes
  10. Proof of Theorem \ref{['thm_mainCode']}
  11. Applications
  12. Linear recurring sequence subgroups
  13. The Schmidt-White conjecture
  14. The asymptotic behavior of the density of non-standard pairs
  15. Concluding remarks

Key Result

Theorem 1.1

Let $C$ be a non-degenerate irreducible cyclic code of length $n$ over ${\mathbb F}_q$, where we assume that $\gcd(n,q)=1$. If $C$ is non-standard, then $C$ is one of the cyclic codes in Examples exa_repetition-exa_imp, or it can be obtained from one of the codes in Examples exa_repetition-exa_imp b

Theorems & Definitions (66)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 56 more