Problems of Information Transmission, Vol. 38, No. 1, 2002, pp. 41–49. Translated from Problemy Peredachi Informatsii, No. 1, 2002, pp. 48–58.
Original Russian Text Copyright
2002 by Handlery, Johannesson, Zyablov.
Encoder and Distance Properties
of Woven Convolutional Codes with One
Tailbiting Component Code
M. Handlery, R. Johannesson, and V. V. Zyablov
Received May 22, 2001
Abstract—Woven convolutional codes with one tailbiting component code are studied and
their generator matrices are given. It is shown that, if the constituent encoders are identical,
a woven convolutional encoder with an outer convolutional warp and one inner tailbiting en-
coder (WIT) generates the same code as a woven convolutional encoder with one outer tailbiting
encoder and an inner convolutional warp (WOT). However, for rate R
< 1 tailbiting encoders,
the WOT cannot be an encoder realization with a minimum number of delay elements. Lower
bounds on the free distance and active distances of woven convolutional codes with a tailbiting
component code are given. These bounds are equal to those for woven codes consisting exclu-
sively of unterminated convolutional codes. However, for woven convolutional codes with one
tailbiting component code, the conditions for the bounds to hold are less strict.
Tailbiting codes can be obtained by terminating convolutional codes into block codes . A rate
R = b/c tailbiting encoder is a rate R = b/c convolutional encoder, which starts in the same state
where it, after encoding b information bits, will end. The parameter is called the tailbiting
length. The generator matrix of a tailbiting code can be described by a binary b × c matrix.
Tailbiting codes often have minimum distances as large as the best linear block codes . Their
error-correcting capability is closely related to that of convolutional codes .
By combining several constituent convolutional codes, we can design a class of convolutional
codes, viz., woven convolutional codes, with large free distances . For certain applications, such
as frequency hopping and OFDM, it is essential to make decisions in subblocks within information
blocks. Then the use of a tailbiting component code in the woven convolutional code construction
is an interesting alternative since the tailbiting component code, being a block code, implicates
decoding of the received symbols within subblocks. This paper investigates encoder properties of
such concatenated codes. We also give lower bounds on the free distance of woven convolutional
codes with one tailbiting component code.
In order to specify the error-correcting capability of a convolutional code beyond what the free
distance predicts, active distances  are considered, which take the sparseness of error patterns
into account. We derive lower bounds on the active distances for woven convolutional codes with
one tailbiting component code. These lower bounds as well as the lower bounds on the free distance
are compared with bounds obtained for the original woven convolutional codes .
In Section 2, two woven code constructions are presented and their generator matrices are
obtained. Basic distance properties of convolutional codes and tailbiting codes are reviewed in
Section 3. In Section 4, we present lower bounds on the free distance and active distances of woven
Supported in part by the Swedish Academy of Science in cooperation with the Russian Academy of Sciences
and in part by the Swedish Research Council for Engineering Sciences, Grant no. 98-501.
2002 MAIK “Nauka/Interperiodica”