Error-Correcting Codes in Projective Spaces via Rank-Metric Codes and Ferrers Diagrams
Abstract
Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant dimension code. The union of these codes is our final constant dimension code. In particular the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant weight codes and the rank-metric codes. Finally, we use puncturing on our final constant dimension codes to obtain large codes in the projective space which are not constant dimension.
Cite
@article{arxiv.0807.4846,
title = {Error-Correcting Codes in Projective Spaces via Rank-Metric Codes and Ferrers Diagrams},
author = {Tuvi Etzion and Natalia Silberstein},
journal= {arXiv preprint arXiv:0807.4846},
year = {2009}
}
Comments
Revised for IEEE Transactions on Information Theory