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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| Published in: | IEEE transactions on information theory Vol. 69; no. 6; pp. 3759 - 3774 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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IEEE
01.06.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Loeliger, Hans-Andrea Yu, Jiun-Hung |
| Author_xml | – sequence: 1 givenname: Jiun-Hung orcidid: 0000-0002-1124-7346 surname: Yu fullname: Yu, Jiun-Hung email: yuji@nycu.edu.tw organization: Department of Electronics and Electrical Engineering, Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu, Taiwan – sequence: 2 givenname: Hans-Andrea orcidid: 0000-0001-7153-7145 surname: Loeliger fullname: Loeliger, Hans-Andrea organization: Department of Information Technology and Electrical Engineering, ETH Zürich, Zürich, Switzerland |
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| SubjectTerms | 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 |
| Title | The Partial-Inverse Approach to Linearized Polynomials and Gabidulin Codes With Applications to Network Coding |
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