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Group Codes

  • Santos González
  • Victor Markov
  • Olga Markova
  • Consuelo MartínezEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1133)

Abstract

Mathematical objects in this paper are group codes. In the first part of the paper we present a survey with some of the main results about group codes, mainly the existence of group codes that are not abelian group codes, the minimal length and the minimal dimension of such codes and the existence of a non-abelian group code that has better parameters than any abelian group code. In particular, in a previous paper [1], we have shown that the minimal dimension of a group code that is not abelian group code is 4. However, all known examples of group codes of dimension 4 that are non-abelian group codes are constructed using groups that are not p-groups.

We do not know if such codes exist for the case of p-groups, but in the second part of this paper we prove that, under some restrictions on the base field, all four-dimensional G-codes for an arbitrary finite p-group G are abelian.

Keywords

Group codes Length Dimension Groups Non-abelian groups Abelian groups 

Notes

Acknowledgements

Authors want specially remember Victor Markov and acknowledge his valuable contribution to this paper before he passed away in July 2019.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Santos González
    • 1
  • Victor Markov
    • 2
  • Olga Markova
    • 2
  • Consuelo Martínez
    • 1
    Email author
  1. 1.Department of MathematicsOviedo UniversityOviedoSpain
  2. 2.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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