  In coding theory, Reed-Solomon codes are one of the most well-known and
widely used classes of error-correcting codes. In this thesis we study and
compare two major strategies known for their decoding procedure, the
Peterson-Gorenstein-Zierler (PGZ) and the Berlekamp-Massey (BM) decoder, in
order to improve existing decoding algorithms and propose faster new ones. In
particular we study a modified version of the PGZ decoder, which we will call
the fast Peterson-Gorenstein-Zierler (fPGZ) decoding algorithm. This
improvement was presented in 1997 by exploiting the Hankel structure of the
syndrome matrix. In this thesis we show that the fPGZ decoding algorithm can be
seen as a particular case of the BM one. Indeed we prove that the intermediate
outcomes obtained in the implementation of fPGZ are a subset of those of the BM
decoding algorithm. In this way, we also uncover the existing relationship
between the leading principal minors of syndrome matrix and the discrepancies
computed by the BM algorithm. Finally, thanks to the study done on the
structure of the syndrome matrix and its leading principal minors, we improve
the error value computation in both the decoding strategies studied
(specifically we prove new error value formulas for the fPGZ and the BM
decoding algorithm) and moreover we state a new iterative formulation of the
PGZ decoder well suited to a parallel implementation on integrated microchips.
Thus using techniques of linear algebra we obtain a parallel decoding algorithm
for Reed-Solomon codes with an O(e) computational time complexity, where e is
the number of errors which occurred, although a fairly large number of
elementary circuit elements is needed.
