Partial Inverses mod m(x) and Reverse Berlekamp–Massey Decoding

This semi-tutorial paper introduces the partial-inverse problem for polynomials and develops its application to decoding Reed-Solomon codes and some related codes. The most natural algorithm to solve the partial-inverse problem is very similar to, but more general than, the Berlekamp-Massey algorith...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 62; no. 12; pp. 6737 - 6756
Main Authors: Jiun-Hung Yu, Loeliger, Hans-Andrea
Format: Journal Article
Language:English
Published: New York IEEE 01.12.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Approximation algorithms
Berlekamp–Massey algorithm
Decoding
Euclidean algorithm
Indexes
Interpolation
Inverse problems
Kernel
key equation
partial-inverse algorithm
partial-inverse problem
polynomial remainder codes
Polynomials
Reed-Solomon codes
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:This semi-tutorial paper introduces the partial-inverse problem for polynomials and develops its application to decoding Reed-Solomon codes and some related codes. The most natural algorithm to solve the partial-inverse problem is very similar to, but more general than, the Berlekamp-Massey algorithm. Two additional algorithms are obtained as easy variations of the basic algorithm: the first variation is entirely new, while the second variation may be viewed as a version of the Euclidean algorithm. Decoding Reed-Solomon codes (and some related codes) can be reduced to the partial-inverse problem, both via the standard key equation and, more naturally, via an alternative key equation with a new converse. Shortened and singly-extended Reed-Solomon codes are automatically included. Using the properties of the partial-inverse problem, two further key equations with attractive properties are obtained. The paper also points out a variety of options for interpolation.
Bibliography:SourceType-Scholarly Journals-1
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2016.2613559