## Title data

Kiermaier, Michael ; Zwanzger, Johannes:

**New ring-linear codes from dualization in projective Hjelmslev geometries.**

*In:* Designs, Codes and Cryptography.
Vol. 66
(2013)
Issue 1–3
.
- pp. 39-55.

ISSN 1573-7586

DOI: https://doi.org/10.1007/s10623-012-9650-1

## Project information

Project title: |
Project's official title Project's id Konstruktive Methoden in der algebraischen Codierungstheorie für lineare Codes über endlichen Kettenringen WA 1666/4 |
---|---|

Project financing: |
Deutsche Forschungsgemeinschaft |

## Abstract in another language

In this article, several new constructions for ring-linear codes are given. The class of base rings are the Galois rings of characteristic 4, which include ℤ₄ as its smallest and most important member. Associated with these rings are the Hjelmslev geometries, and the central tool for the construction is geometric dualization. Applying it to the ℤ₄-preimages of the Kerdock codes and a related family of codes we will call Teichmüller codes, we get two new infinite series of codes and compute their symmetrized weight enumerators. In some cases, residuals of the original code give further interesting codes. The generalized Gray map translates our codes into ordinary, generally non-linear codes in the Hamming space. The obtained parameters include (58, 2⁷, 28)₂, (60, 2⁸, 28)₂, (114, 2⁸, 56)₂, (372, 2¹⁰, 184)₂ and (1988, 2¹², 992)₂ which provably have higher minimum distance than any linear code of equal length and cardinality over an alphabet of the same size (better-than-linear, BTL), as well as (180, 2⁹, 88)₂, (244, 2⁹, 120)₂, (484, 2¹⁰, 240)₂ and (504, 4⁶, 376)₄ where no comparable (in the above sense) linear code is known (better-than-known-linear, BTKL).

## Further data

Item Type: | Article in a journal |
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Refereed: | Yes |

Keywords: | ring-linear code; Kerdock code; Lee weight; homogeneous weight; Galois ring; Gray map; Hjelmslev geometry |

Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics |

Result of work at the UBT: | Yes |

DDC Subjects: | 500 Science > 510 Mathematics |

Date Deposited: | 20 Nov 2014 08:00 |

Last Modified: | 23 Nov 2022 07:36 |

URI: | https://eref.uni-bayreuth.de/id/eprint/3727 |