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—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… (More)

—A new lower bound on the minimum distance of q-ary cyclic codes is proposed. This bound improves upon the Bose– Chaudhuri–Hocquenghem (BCH) bound and, for some codes, upon the Hartmann–Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time… (More)

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… (More)

(Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is considered. This construction requires a modified rank metric, the so–called sum rank metric. For the sum rank metric, the… (More)

Feng and Tzeng's generalization of the Extended Euclidean Algorithm synthesizes the shortest–length linear feedback shift–register for s ≥ 1 sequences, where each sequence has the same length n. In this contribution, it is shown that Feng and Tzeng's algorithm which solves this multi–sequence shift–register problem has time complexity O(sn 2). An… (More)

A new probabilistic decoding algorithm for low-rate Interleaved Reed– Solomon (IRS) codes is presented. This approach increases the error correcting capability of IRS codes compared to other known approaches (e.g. joint decoding) with high probability. It is a generalization of well-known decoding approaches and its complexity is quadratic with the length… (More)

This paper considers fast algorithms for operations on linearized polynomials. Among these results are fast algorithms for division of linearized polynomials, q-transform, multi-point evaluation, computing minimal subspace polynomials and interpolation. The complexity of all these operations is now sub-quadratic in the q-degree of the linearized… (More)

Gabidulin codes can be seen as the rank-metric equivalent of Reed–Solomon codes. It was recently proven, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this… (More)