## Title data

Kiermaier, Michael:

**Codes from translation schemes on Galois rings of characteristic 4.**

*In:* Electronic Notes in Discrete Mathematics.
Vol. 40
(2013)
.
- pp. 175-180.

ISSN 1571-0653

DOI: https://doi.org/10.1016/j.endm.2013.05.032

## 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 |
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Project financing: |
Deutsche Forschungsgemeinschaft |

## Abstract in another language

In [4], it has been shown that the Teichmüller point set in the projective Hjelmslev geometry PHG(Rᵏ) over a Galois ring R of characteristic 4 with k odd is a two-intersection set. From this result, the parameters of the generated codes can be derived, see [8, Fact 5.2]. The resulting Teichmüller Codes have a high minimum distance.

The key step in the proof of the two-weight property in [4] is to show that for a certain supergroup In [4], it has been shown that the Teichmüller point set in the projective Hjelmslev geometry PHG(Rᵏ) over a Galois ring R of characteristic 4 with k odd is a two-intersection set. From this result, the parameters of the generated codes can be derived, see [8, Fact5.2]. The resulting Teichmüller Codes have a high minimum distance. of the Teichmüller units T in a Galois ring S of characteristic 4, the partition A_Σ = {{0},2S^*, Σ, S^*\Σ} induces a translation scheme on (S,+). We generalize these results by characterizing all supergroups Σ of T such that A_Σ induces a symmetric translation scheme. In turn, we get new two-intersection sets in projective Hjelmslev geometries and two new series T_q,k,s and U_q,k,s of R-linear codes. The series T_q,k,s generalizes the Teichmüller codes (special case s=0). The codes U_q,k,s are homogeneous two-weight codes. Application of the dualization construction to T_q,k,s yields another series T^*_q,k,s. The Gray images of the codes T_q,k,s and T^*_q,k,s have a higher minimum distance than all known F_q-linear codes of the same length and size.

## Further data

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

Additional notes: | This article belongs to a special issue: Combinatorics 2012 ed. by
Giorgio Faina |

Keywords: | ring-linear code; Teichmüller group; association scheme; homogeneous weight; projective 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 07:58 |

Last Modified: | 09 Sep 2022 09:09 |

URI: | https://eref.uni-bayreuth.de/id/eprint/3725 |