A Decoding Approach to Reed-Solomon Codes from Their Definition
Because of their importance in applications and their quite simple definition, Reed-Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed-Solomon codes are far from being simple and it is difficult to fit them in introductory courses for un...
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| Vydáno v: | The American mathematical monthly Ročník 125; číslo 4; s. 320 - 338 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Washington
Taylor & Francis
21.04.2018
Taylor & Francis, Ltd Taylor & Francis Ltd |
| Témata: | |
| ISSN: | 0002-9890, 1930-0972 |
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| Abstract | Because of their importance in applications and their quite simple definition, Reed-Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed-Solomon codes are far from being simple and it is difficult to fit them in introductory courses for undergraduates. We introduce a new decoding approach, in a self-contained presentation, which we think may be appropriate for introducing error correction of Reed-Solomon codes to nonexperts. In particular, we interpret Reed-Solomon codes by means of the degree of the interpolation polynomial of the code words and from this derive a decoding algorithm. Compared to the classical algorithms, our algorithm appears to arise more naturally from definitions and to be easier to understand. It is related to the Peterson-Gorenstein-Zierler algorithm (see [
10
] and [
20
]). |
|---|---|
| AbstractList | Because of their importance in applications and their quite simple definition, Reed-Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed-Solomon codes are far from being simple and it is difficult to fit them in introductory courses for undergraduates. We introduce a new decoding approach, in a self-contained presentation, which we think may be appropriate for introducing error correction of Reed-Solomon codes to nonexperts. In particular, we interpret Reed-Solomon codes by means of the degree of the interpolation polynomial of the code words and from this derive a decoding algorithm. Compared to the classical algorithms, our algorithm appears to arise more naturally from definitions and to be easier to understand. It is related to the Peterson-Gorenstein-Zierler algorithm (see [
10
] and [
20
]). Because of their importance in applications and their quite simple definition, Reed–Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed–Solomon codes are far from being simple and it is difficult to fit them in introductory courses for undergraduates. We introduce a new decoding approach, in a self-contained presentation, which we think may be appropriate for introducing error correction of Reed–Solomon codes to nonexperts. In particular, we interpret Reed–Solomon codes by means of the degree of the interpolation polynomial of the code words and from this derive a decoding algorithm. Compared to the classical algorithms, our algorithm appears to arise more naturally from definitions and to be easier to understand. It is related to the Peterson–Gorenstein–Zierler algorithm (see [10] and [20]). Because of their importance in applications and their quite simple definition, Reed– Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed–Solomon codes are far from being simple and it is difficult to fit them in introductory courses for undergraduates. We introduce a new decoding approach, in a selfcontained presentation, which we think may be appropriate for introducing error correction of Reed–Solomon codes to nonexperts. In particular, we interpret Reed–Solomon codes bymeans of the degree of the interpolation polynomial of the code words and from this derive a decoding algorithm. Compared to the classical algorithms, our algorithm appears to arise more naturally from definitions and to be easier to understand. It is related to the Peterson–Gorenstein–Zierler algorithm (see [10] and [20]). |
| Author | Bras-Amorós, Maria |
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| Cites_doi | 10.1137/0109020 10.1109/TIT.1982.1056591 10.1109/TIT.1969.1054260 10.1017/CBO9780511808968 10.1016/0012-365X(92)90567-Y 10.1006/jcom.1997.0439 10.4171/001 10.1109/18.887869 10.1016/0378-3758(95)00088-7 10.1137/0108018 10.1109/PROC.1980.11696 10.1016/S0019-9958(75)90090-X 10.1109/TIT.1987.1057299 10.1109/TIT.2013.2274454 10.1109/TIT.1960.1057586 10.1049/el:19930689 10.1109/TIT.2004.842738 10.1007/s10623-012-9626-1 10.1109/18.490539 10.1109/18.782097 10.1002/j.1538-7305.1948.tb01338.x |
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| References | Peterson W. W. (cit0020) 1960; 6 Sugiyama Y. (cit0026) 1975; 27 Heydtmann A. E. (cit0012) 2000; 46 Shannon C. E. (cit0024) 1948; 27 Guruswami V. (cit0011) 1999; 45 Chambers W. G. (cit0007) 1993; 29 MacWilliams F. J. (cit0014) 1977; 16 Gorenstein D. (cit0010) 1961; 9 Mateer T. D. (cit0017) 2013; 69 Pellikaan R. (cit0018) 1992; 106 Bierbrauer J. (cit0005) 2005 Pellikaan R. (cit0019) 1996; 51 Dornstetter J.-L. (cit0008) 1987; 33 Bossert M. (cit0006) 2013; 59 Berlekamp E. R. (cit0004) 1996; 42 Berlekamp E. R. (cit0003) 1982; 28 Sudan M. (cit0025) 1997; 13 Fedorenko S. V. (cit0009) 2005; 51 Justesen J. (cit0013) 2004 Massey J. L. (cit0015) 1969; 15 Mateer T. D. (cit0016) 2011 Berlekamp E. R. (cit0001) 1968 Berlekamp E. R. (cit0002) 1980; 68 Roman S. (cit0022) 1997 Reed I. S. (cit0021) 1960; 8 Roth R. M. (cit0023) 2006 |
| References_xml | – volume: 9 start-page: 207 year: 1961 ident: cit0010 publication-title: J. Soc. Indust. Appl. Math doi: 10.1137/0109020 – volume: 28 start-page: 869 year: 1982 ident: cit0003 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.1982.1056591 – volume: 15 start-page: 122 year: 1969 ident: cit0015 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.1969.1054260 – volume-title: Introduction to Coding Theory year: 2006 ident: cit0023 doi: 10.1017/CBO9780511808968 – volume-title: Introduction to Coding and Information Theory year: 1997 ident: cit0022 – volume: 106 start-page: 369 year: 1992 ident: cit0018 publication-title: Discrete Math doi: 10.1016/0012-365X(92)90567-Y – volume-title: Algebraic Coding Theory year: 1968 ident: cit0001 – volume: 13 start-page: 180 year: 1997 ident: cit0025 publication-title: J. Complex doi: 10.1006/jcom.1997.0439 – volume-title: A Course in Error-Correcting Codes year: 2004 ident: cit0013 doi: 10.4171/001 – volume: 16 volume-title: The Theory of Error-Correcting Codes year: 1977 ident: cit0014 – volume: 46 start-page: 2614 year: 2000 ident: cit0012 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/18.887869 – volume: 51 start-page: 229 year: 1996 ident: cit0019 publication-title: J. Statist. Plann. Inference doi: 10.1016/0378-3758(95)00088-7 – volume: 8 start-page: 300 year: 1960 ident: cit0021 publication-title: J. Soc. Indust. Appl. Math doi: 10.1137/0108018 – volume: 68 start-page: 564 year: 1980 ident: cit0002 publication-title: Proceedings of the IEEE doi: 10.1109/PROC.1980.11696 – volume: 27 start-page: 87 year: 1975 ident: cit0026 publication-title: Inform. Control doi: 10.1016/S0019-9958(75)90090-X – volume: 33 start-page: 428 year: 1987 ident: cit0008 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.1987.1057299 – volume: 59 start-page: 7320 year: 2013 ident: cit0006 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.2013.2274454 – volume: 6 start-page: 459 year: 1960 ident: cit0020 publication-title: IRE Trans. Inform. Theory doi: 10.1109/TIT.1960.1057586 – volume: 29 start-page: 1031 year: 1993 ident: cit0007 publication-title: Electron. Lett. doi: 10.1049/el:19930689 – volume: 51 start-page: 1196 year: 2005 ident: cit0009 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.2004.842738 – volume: 69 start-page: 107 year: 2013 ident: cit0017 publication-title: Des. Codes Cryptogr doi: 10.1007/s10623-012-9626-1 – volume-title: Introduction to Coding Theory year: 2005 ident: cit0005 – volume: 42 start-page: 704 year: 1996 ident: cit0004 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/18.490539 – volume: 45 start-page: 1757 year: 1999 ident: cit0011 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/18.782097 – volume: 27 start-page: 379 year: 1948 ident: cit0024 publication-title: Bell System Tech. J doi: 10.1002/j.1538-7305.1948.tb01338.x – start-page: 139 volume-title: Cheng, J., Schober, R., chairs. 12th Canadian Workshop on Information Theory (CWIT), 2011. Piscataway, NJ: IEEE year: 2011 ident: cit0016 |
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| Snippet | Because of their importance in applications and their quite simple definition, Reed-Solomon codes can be explained in any introductory course on coding theory.... Because of their importance in applications and their quite simple definition, Reed–Solomon codes can be explained in any introductory course on coding theory.... Because of their importance in applications and their quite simple definition, Reed– Solomon codes can be explained in any introductory course on coding... |
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| SubjectTerms | Algorithms Coding theory Decoding Error correction Interpolation Introductory Courses Polynomials Primary 12Y05 Secondary 94B35 |
| Title | A Decoding Approach to Reed-Solomon Codes from Their Definition |
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