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...with quotient m(x) if g(x) ◦m(x) = g(m(x)) = f(x). Efficient algorithms for all these operations (left and right symbolic multiplication and division) exist and can be found e.g. in [5]. Lemma 1 (cf. =-=[7]-=- Thm. 3.50). Let f(x) ∈ Lq(x, qm) and Fqs be the smallest extension field of Fqm that contains all roots of f(x). Then the set of all roots of f(x) forms a Fq-linear vector space in Fqs . Lemma 2 ([7]...

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...ing metric, but also achieve the Singleton bound with respect to the rank metric and are thus MRD codes. There has been a rising interest in the last decade due to their application in network coding =-=[5]-=-, [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13],...

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...ent fields of coding theory, e.g. in (random) linear network coding [19], space-time coding [10], crisscoss error correction [14] and distributed storage [17]. They were first derived by Gabidulin in =-=[3]-=- and independently by Delsarte in [2]. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polynomials instead of arbitrary polynomials over the finite field Fq (where q i...

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...ed word with respect to the rank metric. I. INTRODUCTION Gabidulin codes are a family of optimal rank-metric codes, useful in different fields of coding theory, e.g. in (random) linear network coding =-=[19]-=-, space-time coding [10], crisscoss error correction [14] and distributed storage [17]. They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These codes can be seen as the...

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...Gabidulin codes are a family of optimal rank-metric codes, useful in different fields of coding theory, e.g. in (random) linear network coding [19], space-time coding [10], crisscoss error correction =-=[14]-=- and distributed storage [17]. They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polyno...

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...(random) linear network coding [19], space-time coding [10], crisscoss error correction [14] and distributed storage [17]. They were first derived by Gabidulin in [3] and independently by Delsarte in =-=[2]-=-. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polynomials instead of arbitrary polynomials over the finite field Fq (where q is a prime power). They are optimal in...

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...4. The previous theorem states that the roots of D(x) form a vector space of degree t which is equal to the span of e1, . . . , en. This is why D(x) is also called the error span polynomial (cf. e.g. =-=[18]-=-). The analogy in the classical Hamming metric set-up is the error locator polynomial, whose roots indicate the locations of the errors, and whose degree equals the number of errors. The interpolation...

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...s defined to be of the form f(x) = n∑ i=0 aix [i] , ai ∈ Fqm , where n is called the q-degree of f(x), assuming that an 6= 0, denoted by qdeg(f). This class of polynomials was first studied by Ore in =-=[12]-=-. One can easily check that f(x1 + x2) = f(x1) + f(x2) and f(λx1) = λf(x1) for any x1, x2 ∈ Fqm and λ ∈ Fq , hence the name linearized. The set of all q-linearized polynomials over Fqm is denoted by L...

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...coding [5], [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], =-=[9]-=-, [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabi...

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... then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], =-=[20]-=-, whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found...

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...g [5], [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], =-=[13]-=-, [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin ...

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...work coding [5], [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], =-=[4]-=-, [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted...

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...ue decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], =-=[11]-=-, [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], p...

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...this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present a parametric approach analogous to the one for list decoding Reed-Solomon codes from =-=[1]-=-. In a similar way as [9] we use interpolation, however unlike [9] we perform list decoding rather than unique decoding. A difference between our paper and the papers [9], [23] is that our approach is...

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... unique decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. =-=[8]-=-, [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [...

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... Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13], [15], =-=[16]-=-, [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be...

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...oding radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], =-=[22]-=-, [23]. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present...

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...ereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes can be found in =-=[21]-=-. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present a parametric approach analogous to the one for list decoding Reed-Solomon codes ...

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...) requirement 2) in Theorem 11. In fact, it can be proven that G is a minimal Gröbner basis for the interpolation module and has the so-called Predictable Leading Monomial Property analogous to [1], =-=[6]-=-. As a result of this property, the elements in the two for-loops that fulfill the divisibility requirement correspond to codewords with rank distance t = j−ℓ2+k−1 from r. Due to space limitations we ...

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... [19]. Since then a lot of work has been done on how to decode these codes. The question of minimum distance decoding inside the unique decoding radius has been addressed e.g. in [3], [4], [9], [13], =-=[15]-=-, [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], [23]. Related work on list-decoding lifted Gabidulin codes ...

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...radius has been addressed e.g. in [3], [4], [9], [13], [15], [16], [20], whereas the more general setting of list decoding, beyond the unique decoding radius, is investigated in e.g. [8], [11], [22], =-=[23]-=-. Related work on list-decoding lifted Gabidulin codes can be found in [21]. In this work we explore list decoding further and, in contrast to the Sudan-Guruswami approach of [11], [22], present a par...

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...of optimal rank-metric codes, useful in different fields of coding theory, e.g. in (random) linear network coding [19], space-time coding [10], crisscoss error correction [14] and distributed storage =-=[17]-=-. They were first derived by Gabidulin in [3] and independently by Delsarte in [2]. These codes can be seen as the q-analog of ReedSolomon codes, using q-linearized polynomials instead of arbitrary po...