Decoding Interleaved Gabidulin Codes using Alekhnovich's Algorithm

Puchinger, Sven; Müelich, Sven; Mödinger, David; Rosenkilde, Johan; Bossert, Martin

doi:10.1016/j.endm.2017.02.029

Cited by 9 publications

(18 citation statements)

References 7 publications

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“…Based on a preprint of this paper, in [39] it is shown how to further reduce the complexity for decoding Interleaved Gabidulin codes using a divide-&-conquer version of Algorithm 3, matching the complexity of [43].…”

Section: Resultsmentioning

confidence: 99%

“…These match the best known algorithms for these applications but solve more general problems, and it demonstrates that the row reduction methodology is both flexible and fast for skew polynomial rings. Building on this paper, [39] proposes an algorithm which improves upon the best known complexity for decoding Interleaved Gabidulin codes. Section 1.1 summarizes related work.…”

Section: Introductionmentioning

confidence: 99%

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Row reduction applied to decoding of rank-metric and subspace codes

Puchinger

Rosenkilde

et al. 2016

Des. Codes Cryptogr.

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For many algebraic codes the main part of decoding can be reduced to row reduction of a module basis, enabling general, flexible and highly efficient algorithms. Inspired by this, we develop an approach of transforming matrices over skew polynomial rings into certain normal forms. We apply this to solve generalised shift register problems, or Padé approximations, over skew polynomial rings which occur in error and erasure decoding -Interleaved Gabidulin codes. We obtain an algorithm with complexity O( µ 2 ) where µ measures the size of the input problem. Further, we show how to listdecode Mahdavifar-Vardy subspace codes in O( r 2 m 2 ) time, where m is a parameter proportional to the dimension of the codewords' ambient space and r is the dimension of the received subspace.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Row reduction applied to decoding of rank-metric and subspace codes

Puchinger

Rosenkilde

et al. 2016

Des. Codes Cryptogr.

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“…KR3517/9-1. rithm in [9]. Both algorithms have complexity O ∼ ( 3 M(n)), where O ∼ neglects logarithmic factors and M(n) is the complexity of multiplying two linearized polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…The latter interpretation is an advantage over the other known decoders. As any other interpolationbased decoder, the algorithm consists of two steps: The interpolation step can be implemented with complexity O( 2 n 2 ) over F q m using the algorithm in [13] or with O ∼ ( 3 M( n)) using the Alekhnovich-like algorithm in [9] (see also [14] for more details). The current bottleneck of the decoder (in asymptotic dependence on n) is the root-finding step, for which there exist two algorithms: the method in [13] computes an affine basis of the root space in O( 3 n 2 ) over F q m and the one in [15], [16] has complexity O( 2 n 2 ).…”

Section: Introductionmentioning

confidence: 99%

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Fast Root Finding for Interpolation-Based Decoding of Interleaved Gabidulin Codes

Bartz

Jerkovits

Puchinger

et al. 2019

2019 IEEE Information Theory Workshop (ITW)

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We show that the root-finding step in interpolationbased decoding of interleaved Gabidulin codes can be solved by finding a so-called minimal approximant basis of a matrix over a linearized polynomial ring. Based on existing fast algorithms for computing such bases over ordinary polynomial rings, we develop fast algorithms for computing them over linearized polynomials. As a result, root finding costs O ∼ (ω M(n)) operations in Fqm , where is the interleaving degree, n the code length, Fqm the base field of the code, 2 ≤ ω ≤ 3 the matrix multiplication exponent, and M(n) ∈ O(n 1.635) is the complexity of multiplying two linearized polynomials of degree at most n. This is an asymptotic improvement upon the previously fastest algorithm of complexity O(3 n 2), in some cases O(2 n 2).

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Fast decoding of lifted interleaved linearized Reed–Solomon codes for multishot network coding

Bartz,

Puchinger

2024

Des. Codes Cryptogr.

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Martínez-Peñas and Kschischang (IEEE Trans. Inf. Theory 65(8):4785–4803, 2019) proposed lifted linearized Reed–Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed–Solomon (LILRS) codes. Compared to the construction by Martínez-Peñas–Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for LILRS that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique Loidreau–Overbeck-like decoder for LILRS codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations.

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Decoding Interleaved Gabidulin Codes using Alekhnovich's Algorithm

Cited by 9 publications

References 7 publications

Row reduction applied to decoding of rank-metric and subspace codes

Row reduction applied to decoding of rank-metric and subspace codes

Fast Root Finding for Interpolation-Based Decoding of Interleaved Gabidulin Codes

Fast decoding of lifted interleaved linearized Reed–Solomon codes for multishot network coding

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