Non-free extensions of the simplex codes over a chain ring with four elements (original) (raw)

Abstract

Let R be a chain ring with four elements. In this paper, we present two new constructions of _R_-linear codes that contain a subcode associated with a simplex code over the ring R. The simplex codes are defined as the codes generated by a matrix having as columns the homogeneous coordinates of all points in some projective Hjelmslev geometry PHG(R k). The first construction generalizes a recent result by Kiermaier and Zwanzger to codes of arbitrary dimension. We provide a geometric interpretation of their construction which is then extended to projective Hjelmslev spaces of arbitrary dimension. The second construction exploits the possibility of adding two non-free rows to the generator matrix of a linear code over R associated with a given point set. Though the construction works over both chain rings with four elements, the better codes are obtained for \({R=\mathbb{Z}_4}\) .

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Authors and Affiliations

  1. Department of Information Science and Electronics Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China
    Thomas Honold
  2. New Bulgarian University, 21 Montevideo Str., 1618, Sofia, Bulgaria
    Ivan Landjev

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  1. Thomas Honold
  2. Ivan Landjev

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Correspondence toIvan Landjev.

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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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Honold, T., Landjev, I. Non-free extensions of the simplex codes over a chain ring with four elements.Des. Codes Cryptogr. 66, 27–38 (2013). https://doi.org/10.1007/s10623-012-9649-7

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