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New ring-linear codes from dualization in projective Hjelmslev geometries

Titelangaben

Kiermaier, Michael ; Zwanzger, Johannes:
New ring-linear codes from dualization in projective Hjelmslev geometries.
In: Designs, Codes and Cryptography. Bd. 66 (Januar 2013) Heft 1–3 . - S. 39-55.
ISSN 0925-1022
DOI: 10.1007/s10623-012-9650-1

Angaben zu Projekten

Projekttitel:
Offizieller ProjekttitelProjekt-ID
Konstruktive Methoden in der algebraischen Codierungstheorie für lineare Codes über endlichen Kettenringen.WA 1666/4

Projektfinanzierung: Deutsche Forschungsgemeinschaft

Abstract

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).

Weitere Angaben

Publikationsform: Artikel in einer Zeitschrift
Begutachteter Beitrag: Ja
Keywords: ring-linear code; Kerdock code; Lee weight; homogeneous weight; Galois ring; Gray map; Hjelmslev geometry
Institutionen der Universität: Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik II (Computeralgebra)
Fakultäten
Fakultäten > Fakultät für Mathematik, Physik und Informatik
Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut
Titel an der UBT entstanden: Ja
Themengebiete aus DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Eingestellt am: 20 Nov 2014 08:00
Letzte Änderung: 12 Feb 2015 09:34
URI: https://eref.uni-bayreuth.de/id/eprint/3727