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Series in
Signal and Information Processing, Vol. 26
edited by Hans-Andrea Loeliger
Jiun-Hung Yu
A Partial-Inverse Approach to Decoding
Reed-Solomon Codes and
Polynomial Remainder Codes
1.
Auflage/1st edition 2015, XIV, 130 Seiten/pages, € 64,00.
ISBN 978-3-86628-527-9
The thesis develops a new approach to the central themes of algebraic
coding theory. The focus here is the newly introduced concept of a
partial-inverse polynomial in a quotient ring F[x]/m(x). In particular, the
decoding of Reed-Solomon codes can be attributed to the computation of a
partial-inverse polynomial.
The problem of practical computation of a partial-inverse polynomial is
closely related to the problem of shift-register synthesis, which is based on
the well-known Berlekamp-Massey algorithm.
A major result of this work is a (new) algorithm for computing a
partial-inverse polynomial. The new algorithm is very similar to the Berlekamp-Massey algorithm, but it is applicable generally,
e.g., to extended Reed-Solomon codes and polynomial
remainder codes. The algorithm can also be easily transformed into the
so-called Euclidean algorithm, and thus provides a new derivation of the later.
For decoding Reed-Solomon codes, the algorithm can be directly applied
to the classical key equation; however, mathematically natural is the
application to a new key equation that applies in particular to generalizations
of Reed-Solomon codes. Two new interpolation are also
presented to accompany this new key equation.
Another focus of this work is the polynomial remainder codes, a natural
generalization of Reed-Solomon codes. The theory of such codes is carefully
constructed as in earlier work. In particular, varying degrees of remainders
are allowed, resulting in two different definitions of the distance between two
codewords. The decoding of these codes leads directly
to the mentioned new key equation.
A focus of the recent algebraic coding theory is the decoding of errors
beyond half the minimum distance. A mainline of such algorithms is based on
generalization of the Berlekamp-Massey algorithm on
several parallel sequences. In this work, a corresponding generalization of the
decoding algorithm via some partial-inverse polynomials is also developed.
Keywords: Error-Correcting Codes, Reed-Solomon Codes, Algebraic Coding Theory,
Polynomial Remainder Codes, Berlekamp-Massey
Algorithm, Padé Approximation, Euclidean Algorithm,
Partial-Inverse Problem, Simultaneous Partial-Inverse Problem.
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