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New ring-linear codes from dualization in projective Hjelmslev geometries

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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 \({\mathbb {Z}_4}\) 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 \({\mathbb {Z}_4}\) -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).

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Michael Kiermaier

  2. Siemens AG, CT T DE IT1, Otto-Hahn-Ring 6, 81739, München, Deutschland

    Johannes Zwanzger

Authors
  1. Michael Kiermaier
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  2. Johannes Zwanzger
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Correspondence to Michael Kiermaier.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Kiermaier, M., Zwanzger, J. New ring-linear codes from dualization in projective Hjelmslev geometries. Des. Codes Cryptogr. 66, 39–55 (2013). https://doi.org/10.1007/s10623-012-9650-1

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

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