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

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 of Locally Repairable Codes

We show that locally repairable codes (LRCs) can be list decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error correction capabilities. The new decoding radius is derived and the asymptotic behavior is analyzed. We give a general list decoding algorithm for LRCs that achieves this radius along with an explicit realization for a class of LRCs based on Reed-Solomon codes (Tamo-Barg LRCs). Further, a probabilistic algorithm for unique decoding of low complexity is given and its success probability analyzed

Optimal Ferrers Diagram Rank-Metric Codes

Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing optimal codes for different diagrams and parameters.Comment: to be presented in Algebra, Codes, and Networks, Bordeaux, June 16 - 20, 201