ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 4, pp. 324–333.
Pleiades Publishing, Inc., 2012.
Original Russian Text
J.H. Weber, V.R. Sidorenko, C. Senger, K.A.S. Abdel-Ghaﬀar, 2012, published in Problemy Peredachi Informatsii, 2012,
Vol. 48, No. 4, pp. 30–40.
Asymptotic Single-Trial Strategies for GMD Decoding
with Arbitrary Error-Erasure Tradeoﬀ
J. H. Weber
, and K. A. S. Abdel-Ghaﬀar
Delft University of Technology, The Netherlands
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Ulm University, Germany
Ulm University, Germany
University of California, Davis, CA, USA
Received April 23, 2012; in ﬁnal form, October 15, 2012
Abstract—Generalized minimum distance (GMD) decoders allow for combining some virtues
of probabilistic and algebraic decoding approaches at a low complexity. We investigate single-
trial strategies for GMD decoding with arbitrary error-erasure tradeoﬀ, based on either erasing
a fraction of the received symbols or erasing all symbols whose reliability values are below a
certain threshold. The fraction/threshold may be either static or adaptive, where adaptive
means that the erasing is a function of the channel output. Adaptive erasing based on a
threshold is a new technique that has not been investigated before. An asymptotic approach
is used to evaluate the error-correction radius for each strategy. Both known and new results
appear as special cases of this general framework.
Generalized minimum distance (GMD) decoding, as was introduced by Forney [1, 2], permits
ﬂexible use of reliability information in algebraic decoding algorithms for error correction. It applies
to both binary and nonbinary codes. The main idea is to use a simple algebraic error-erasure decoder
in a multi-trial fashion, with diﬀerent erasing patterns based on reliability information from the
channel and termination based on a certain distance criterion. Each erasing pattern is likely to
turn a set of errors (channel disturbance with unknown location) into erasures (channel disturbance
with known location). The latter are generally easier to correct; hence, virtues of probabilistic and
algebraic decoding approaches can be combined.
Many variations on and extensions or improvements to the GMD decoding principle have been
proposed over the years. For example, in [3, 4], the eﬀects of reducing the maximum number of
decoding trials were studied. Further, in , GMD decoding with a bounded distance decoder cor-
rectingupto(d − e − 1)/λ errors was introduced, where d is the code’s Hamming distance, e is the
number of erasures, and λ is a real number from (1, 2] denoted as the error-erasure tradeoﬀ. It spec-
iﬁes the relative cost of correcting errors vs. correcting erasures. In the classical case one has λ =2,
while smaller values of λ are of interest for decoding certain classes of punctured Reed–Solomon
codes as well as for some modern algebraic decoders like the Sudan  and Guruswami–Sudan