The Partial-Inverse Approach to Linearized Polynomials and Gabidulin Codes With Applications to Network Coding

This paper introduces the partial-inverse problem for linearized polynomials and develops its application to decoding Gabidulin codes and lifted Gabidulin codes in linear random network coding. The proposed approach is a natural generalization of its counterpart for ordinary polynomials, thus provid...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 69; no. 6; pp. 3759 - 3774
Main Authors: Yu, Jiun-Hung, Loeliger, Hans-Andrea
Format: Journal Article
Language:English
Published: New York IEEE 01.06.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Berlekamp-Massey algorithm
Codes
Coding
Decoding
Encoding
Euclidean algorithm
Gabidulin codes
Interpolation
Inverse problems
key equation
Linearization
Measurement
Network coding
partial-inverse algorithm
partial-inverse problem
Polynomials
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:This paper introduces the partial-inverse problem for linearized polynomials and develops its application to decoding Gabidulin codes and lifted Gabidulin codes in linear random network coding. The proposed approach is a natural generalization of its counterpart for ordinary polynomials, thus providing a unified perspective on Reed-Solomon codes for the Hamming metric and for the rank metric. The basic algorithm for solving the partial-inverse problem is a common parent algorithm of a Berlekamp-Massey algorithm, a Euclidean algorithm, and yet another algorithm, all of which are obtained as easy variations of the basic algorithm. Decoding Gabidulin codes can be reduced to the partial-inverse problem via a key equation with a new converse. This paper also develops new algorithms for interpolating crisscross erasures and for joint decoding of errors, erasures, and deviations in random network coding.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2023.3236720