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-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

On interpolation-based decoding of a class of maximum rank distance codes

In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.acceptedVersio

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

Encoding and decoding of several optimal rank metric codes

This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to ⌊(d − 1)/2⌋ in polynomial-time, where d is the minimum distance of the code.publishedVersio

A module minimization approach to Gabidulin decoding via interpolation

We focus on iterative interpolation-based decoding of Gabidulin codes and present an algorithm that computes a minimal basis for an interpolation module. We extend existing results for Reed-Solomon codes in showing that this minimal basis gives rise to a parametrization of elements in the module that lead to all Gabidulin decoding solutions that are at a fixed distance from the received word. Our module-theoretic approach strengthens the link between Gabidulin decoding and Reed-Solomon decoding, thus providing a basis for further work into Gabidulin list decoding