Problems of Information Transmission, Vol. 37, No. 3, 2001, pp. 190–205. Translated from Problemy Peredachi Informatsii, No. 3, 2001, pp. 6–23.
Original Russian Text Copyright
2001 by Truhachev, Lentmaier, Zigangirov.
INFORMATION THEORY AND CODING THEORY
Some Results Concerning the Design
and Decoding of Turbo-Codes
D. V. Truhachev, M. Lentmaier, and K. Sh. Zigangirov
Received February 13, 2001; in ﬁnal form, May 22, 2001
Abstract—We consider the following problems related to the construction and analysis of
turbo-codes: asymptotic behavior of interleavers (permutors), asymptotic behavior of the min-
imum distance, and also some examples of practical application of the developed methods to
Turbo-codes, also called parallel concatenated codes, were introduced by a group of French
authors in 1993 . They can be considered as a particular case of low-density parity-check codes .
The main advantage of these codes is that they can be decoded iteratively. While the complexity
of iterative decoding is considerably lower than the complexity of maximum-likelihood decoding
(ML), the reliability is signiﬁcantly higher than for ML decodable codes with the same complexity.
There exist two diﬀerent approaches to the analysis of low-density codes. In the ﬁrst approach,
random ensembles of codes are considered and code characteristics averaged over the whole ensemble
are estimated. This approach is productive for the analysis of general characteristics of codes (for
example, minimum distance or spectrum) and maximum-likelihood decoding. The second approach
is to analyze specially chosen codes. Existence of these codes in the ensemble is proved using special
constructive methods. From our point of view, this way is preferable for the analysis of iterative
decoding of codes. The two approaches were introduced in the work by Gallager  and developed,
in particular, in the papers by Zyablov and Pinsker , Margulis [4–6], Tanner , Benedetto and
Montorsi , and other recently published works.
In this article, we adhere to the second approach and analyze asymptotic properties of iterative
decoding of binary turbo-codes. The decoding comprises iterative calculation of a posteriori prob-
abilities of information symbols. The “correct” calculation of the a posteriori probabilities puts
some constraints on the structure of the turbo-encoder, in particular, on the construction of the
. The main results of this work are a proof of existence of a permutor that
provides “correct” calculation of the a posteriori probabilities of symbols and a proof of existence
of turbo-codes with minimum distance growing logarithmically with length of the code.
Binary turbo-codes with memory-one component encoders are studied in more detail. Although
these codes have worse characteristics than turbo-codes with component encoders with larger mem-
ory, their structure allows for a simple mathematical analysis. The authors are going to carry out
Supported in part by the Swedish Research Council for Engineering Sciences (Grant 98-216).
The term“interleaver” is usually used in the technical literature to describe a device consisting of a table
which is columnwise ﬁlled by input symbols and read row-wise. We call a “permutor” a general device
that changes the order of symbols in an input information sequence. The class of these devices contains
standard interleavers as a particular case. Earlier, we used the term “scrambler” but since this term
speciﬁes other devices in communication theory and cryptography, we decided to change it, following the
advice of J. Massey.
2001 MAIK “Nauka/Interperiodica”