ISSN 0032-9460, Problems of Information Transmission, 2017, Vol. 53, No. 2, pp. 114–135.
Pleiades Publishing, Inc., 2017.
Original Russian Text
I.V. Zhilin, V.V. Zyablov, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 2, pp. 16–39.
Generalized Error-Locating Codes with Component
Codes over the Same Alphabet
I. V. Zhilin
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
Received April 23, 2015; in ﬁnal form, March 7, 2017
Abstract—We consider generalized error locating (GEL) codes over the same alphabet for both
component codes. We propose an algorithm for computing an upper bound on the decoding
error probability under known input symbol error rate and code parameters. Is is used to
construct an algorithm for selecting code parameters to maximize the code rate for a given
construction and given input and output error probabilities. A lower bound on the decoding
error probability is given. Examples of plots of decoding error probability versus input symbol
error rate are given, and their behavior is explained.
Presently, due to limits on frequency spectra and permanently increasing data bandwidth re-
quirements, coded modulation constructions based on modulation with high order becomes more
and more needed in communication systems. At the same time, error-correcting codes that have
been exploited in widely used systems were binary and could be suboptimal for the corresponding
This paper considers a known type of error-correcting codes, namely generalized error locating
(GEL) codes . The main purpose of these codes is getting very low values of the decoding error
Unlike error-locating codes [2–4], which were proposed for error detection and location, GEL
codes can be used for error correction [1, 5, 6]. Equivalence of GEL codes and generalized concate-
nated (GC) codes [8–12] was shown in .
GEL codes are most eﬃcient for channels with relatively low error rates. For example, they were
proposed for ﬁber-optic communication systems [13,14], since they allow working with rather high
rate even under input error rate of 10
. Furthermore, they were suggested to be used in magnetic
data storage systems [15, 16]. The paper  proposes a concatenated code version which uses a
space-time code as an inner code and a GEL code as an outer code; the paper  proposes a GEL
code with short binary codes as inner codes and LDPC codes as outer codes.
A codeword of a GEL code is usually written as a matrix with height equal to the length of the
inner codes and width equal to the length of the outer codes.
GEL codes are a class of generalized concatenated codes in which columns of a codeword are
not codewords of any code (see Section 2). This construction is only eﬃcient in the case where
errors are of such a kind that most columns of received codewords contain no errors. Form this
it is easily seen that for a discrete memoryless symmetric channel, inner codes should be possibly
short to satisfy this condition.