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Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric.

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Bibliographic Details
Title: Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric.
Authors: Bartz, Hannes, Jerkovits, Thomas, Puchinger, Sven, Rosenkilde, Johan
Source: IEEE Transactions on Information Theory; Aug2021, Vol. 67 Issue 8, p5026-5050, 25p
Subject Terms: ALGORITHMS, POLYNOMIAL rings, REED-Solomon codes, POLYNOMIALS, LINEAR network coding, MULTIPLICATION, DECODING algorithms
Abstract: We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed–Solomon codes in the sum-rank metric. The speed-ups are achieved by new algorithms that reduce the cores of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials. [ABSTRACT FROM AUTHOR]
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Database: Complementary Index
Description
Abstract:We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed–Solomon codes in the sum-rank metric. The speed-ups are achieved by new algorithms that reduce the cores of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials. [ABSTRACT FROM AUTHOR]
ISSN:00189448
DOI:10.1109/TIT.2021.3067318