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S

Series in
Signal and Information Processing, Vol. 26
edited by HansAndrea Loeliger
JiunHung Yu
A PartialInverse Approach to Decoding
ReedSolomon Codes and
Polynomial Remainder Codes
1.
Auflage/1st edition 2015, XIV, 130 Seiten/pages, € 64,00.
ISBN 9783866285279
The thesis develops a new approach to the central themes of algebraic
coding theory. The focus here is the newly introduced concept of a
partialinverse polynomial in a quotient ring F[x]/m(x). In particular, the
decoding of ReedSolomon codes can be attributed to the computation of a
partialinverse polynomial.
The problem of practical computation of a partialinverse polynomial is
closely related to the problem of shiftregister synthesis, which is based on
the wellknown BerlekampMassey algorithm.
A major result of this work is a (new) algorithm for computing a
partialinverse polynomial. The new algorithm is very similar to the BerlekampMassey algorithm, but it is applicable generally,
e.g., to extended ReedSolomon codes and polynomial
remainder codes. The algorithm can also be easily transformed into the
socalled Euclidean algorithm, and thus provides a new derivation of the later.
For decoding ReedSolomon 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 ReedSolomon 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 ReedSolomon 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 BerlekampMassey algorithm on
several parallel sequences. In this work, a corresponding generalization of the
decoding algorithm via some partialinverse polynomials is also developed.
Keywords: ErrorCorrecting Codes, ReedSolomon Codes, Algebraic Coding Theory,
Polynomial Remainder Codes, BerlekampMassey
Algorithm, Padé Approximation, Euclidean Algorithm,
PartialInverse Problem, Simultaneous PartialInverse Problem.
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