List and Unique Error-Erasure Decoding of Interleaved Gabidulin Codes with Interpolation Techniques

A new interpolation-based decoding principle for interleaved Gabidulin codes is presented. The approach consists of two steps: First, a multi-variate linearized polynomial is constructed which interpolates the coefficients of the received word and second, the roots of this polynomial have to be found. Due to the specific structure of the interpolation polynomial, both steps (interpolation and root-finding) can be accomplished by solving a linear system of equations. This decoding principle can be applied as a list decoding algorithm (where the list size is not necessarily bounded polynomially) as well as an efficient probabilistic unique decoding algorithm. For the unique decoder, we show a connection to known unique decoding approaches and give an upper bound on the failure probability. Finally, we generalize our approach to incorporate not only errors, but also row and column erasures.Comment: accepted for Designs, Codes and Cryptography; presented in part at WCC 2013, Bergen, Norwa

List-Decoding Gabidulin Codes via Interpolation and the Euclidean Algorithm

We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Euclidean algorithm with respect to composition of polynomials. For a given received word, our decoding algorithm computes a list of all codewords that are closest to the received word with respect to the rank metric.Comment: Submitted to ISITA 2014, IEICE copyright upon acceptanc

Iterative List-Decoding of Gabidulin Codes via Gr\"obner Based Interpolation

We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.Comment: Submitted to IEEE Information Theory Workshop 2014 in Hobart, Australi

List and Probabilistic Unique Decoding of Folded Subspace Codes

A new class of folded subspace codes for noncoherent network coding is presented. The codes can correct insertions and deletions beyond the unique decoding radius for any code rate $R\in[0,1]$. An efficient interpolation-based decoding algorithm for this code construction is given which allows to correct insertions and deletions up to the normalized radius $s(1-((1/h+h)/(h-s+1))R)$, where $h$ is the folding parameter and $s\leq h$ is a decoding parameter. The algorithm serves as a list decoder or as a probabilistic unique decoder that outputs a unique solution with high probability. An upper bound on the average list size of (folded) subspace codes and on the decoding failure probability is derived. A major benefit of the decoding scheme is that it enables probabilistic unique decoding up to the list decoding radius.Comment: 6 pages, 1 figure, accepted for ISIT 201