Row Reduction Applied to Decoding of Rank Metric and Subspace Codes
Abstract
We show that decoding of $\ell$Interleaved Gabidulin codes, as well as list$\ell$ decoding of MahdavifarVardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of $\F[x]$ matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding $\ell$Interleaved Gabidulin codes. We obtain an algorithm with complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem and is proportional to the code length $n$ in the case of decoding. Further, we show how to perform the interpolation step of list$\ell$decoding MahdavifarVardy codes in complexity $O(\ell n^2)$, where $n$ is the number of interpolation constraints.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 arXiv:
 arXiv:1510.04728
 Bibcode:
 2015arXiv151004728P
 Keywords:

 Computer Science  Information Theory
 EPrint:
 Accepted for Designs, Codes and Cryptography