Bounds on List Decoding of Rank-Metric Codes

So far, there is no polynomial-time list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds on the list size of rank-metric codes in order to understand whether polynomial-time list decoding is possible or whether it works only with exponential time complexity. Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomial-time list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rank-metric code of length $n$ and minimum rank distance $d$. The third bound proves that there exists a rank-metric code over \Fqm of length $n \leq m$ such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on $n$ and $d$ similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Information Theory, short version presented at ISIT 201

Bounds on List Decoding Gabidulin Codes

An open question about Gabidulin codes is whether polynomial-time list decoding beyond half the minimum distance is possible or not. In this contribution, we give a lower and an upper bound on the list size, i.e., the number of codewords in a ball around the received word. The lower bound shows that if the radius of this ball is greater than the Johnson radius, this list size can be exponential and hence, no polynomial-time list decoding is possible. The upper bound on the list size uses subspace properties.Comment: Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2012), Pomorie : Bulgaria (2012

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

On the Geometry of Balls in the Grassmannian and List Decoding of Lifted Gabidulin Codes

The finite Grassmannian $\mathcal{G}_{q}(k,n)$ is defined as the set of all $k$-dimensional subspaces of the ambient space $\mathbb{F}_{q}^{n}$. Subsets of the finite Grassmannian are called constant dimension codes and have recently found an application in random network coding. In this setting codewords from $\mathcal{G}_{q}(k,n)$ are sent through a network channel and, since errors may occur during transmission, the received words can possible lie in $\mathcal{G}_{q}(k',n)$, where $k'\neq k$. In this paper, we study the balls in $\mathcal{G}_{q}(k,n)$ with center that is not necessarily in $\mathcal{G}_{q}(k,n)$. We describe the balls with respect to two different metrics, namely the subspace and the injection metric. Moreover, we use two different techniques for describing these balls, one is the Pl\"ucker embedding of $\mathcal{G}_{q}(k,n)$, and the second one is a rational parametrization of the matrix representation of the codewords. With these results, we consider the problem of list decoding a certain family of constant dimension codes, called lifted Gabidulin codes. We describe a way of representing these codes by linear equations in either the matrix representation or a subset of the Pl\"ucker coordinates. The union of these equations and the equations which arise from the description of the ball of a given radius in the Grassmannian describe the list of codewords with distance less than or equal to the given radius from the received word.Comment: To be published in Designs, Codes and Cryptography (Springer

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