Non-free extensions of the simplex codes over a chain ring with four elements

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 p...

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Bibliographic Details
Published in:Designs, codes, and cryptography codes, and cryptography, 2013, Vol.66 (1-3), p.27-38
Main Authors: Honold, Thomas, Landjev, Ivan
Format: Article
Language:English
Subjects:
Circuits
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Information and Communication
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Summary: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 .
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-012-9649-7