Related papers: On deep-holes of Gabidulin codes
Twisted Gabidulin codes are an extension of Gabidulin codes and have recently attracted great attention. In this paper, we study three classes of twisted Gabidulin codes with different twists. Moreover, we establish necessary and sufficient…
In this work we present a new criterion to check if a given rank-metric code is a maximum rank distance (MRD) code. Moreover, we derive a criterion to check if a given MRD code is a generalized Gabidulin code. We then use these results to…
For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole.…
Wachter-Zeh in [42], and later together with Raviv [31], proved that Gabidulin codes cannot be efficiently list decoded for any radius $\tau$, providing that $\tau$ is large enough. Also, they proved that there are infinitely many choices…
Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word $u$ is a deep hole of the standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation…
We address the problem of decoding Gabidulin codes beyond their unique error-correction radius. The complexity of this problem is of importance to assess the security of some rank-metric code-based cryptosystems. We propose an approach that…
For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized Reed-Solomon codes is proved to be…
The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. {Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper,…
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…
This article discusses the decoding of Gabidulin codes and shows how to extend the usual decoder to any supercode of a Gabidulin code at the cost of a significant decrease of the decoding radius. Using this decoder, we provide polynomial…
We study the problem of classifying deep holes of Reed-Solomon codes. We show that this problem is equivalent to the problem of classifying MDS extensions of Reed-Solomon codes by one digit. This equivalence allows us to improve recent…
This paper investigates the decoding of certain Gabidulin codes that were transmitted over a channel with space-symmetric errors. Space-symmetric errors are additive error matrices that have the property that their column and row spaces are…
Maximum rank-distance (MRD) codes are extremal codes in the space of $m\times n$ matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are…
Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the…
In this paper an interpolation-based decoding algorithm to decode Gabidulin codes, transmitted through a finely restricted channel, is proposed. The algorithm is able to decode rank errors beyond half the minimum distance by one unit. Also…
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…
Gabidulin codes, serving as the rank-metric counterpart of Reed-Solomon codes, constitute an important class of maximum rank distance (MRD) codes. However, unlike the fruitful positive results about the list decoding of Reed-Solomon codes,…
Gabidulin codes are the rank-metric analogs of Reed-Solomon codes and have a major role in practical error control for network coding. This paper presents new encoding and decoding algorithms for Gabidulin codes based on low-complexity…
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…
In this paper, we obtain new results on the covering radius and deep holes for projective Reed-Solomon (PRS) codes.