ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 1, pp. 1–14.
Pleiades Publishing, Inc., 2011.
Original Russian Text
I.E. Bocharova, F. Hug, R. Johannesson, B.D. Kudryashov, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47,
No. 1, pp. 3–18.
Woven Convolutional Graph Codes
with Large Free Distances
I. E. Bocharova
, R. Johannesson
, and B. D. Kudryashov
St. Petersburg State University of Information Technologies,
Mechanics and Optics (ITMO)
Lund University, Sweden
Received May 5, 2010; in ﬁnal form, October 19, 2010
Abstract—Constructions of woven graph codes based on constituent convolutional codes are
studied, and examples of woven convolutional graph codes are presented. Existence of codes
satisfying the Costello lower bound on the free distance within a random ensemble of woven
graph codes based on s-partite, s-uniform hypergraphs is shown, where s depends only on the
code rate. Simulation results for Viterbi decoding of woven graph codes are presented and
Woven graph codes are concatenated graph-based codes with constituent block or convolutional
codes [1,2]. A distinguishing feature of these codes is that the length of the constituent code is a
multiple of the underlying graph degree c; i.e., their length is lc,wherel is an integer. In particular,
when l tends to inﬁnity, we obtain woven graph codes with constituent convolutional codes. While,
for example, serial concatenated convolutional codes are obtained by combining generator matrices,
woven graph codes are obtained by combining the corresponding parity-check matrices.
In , existence of codes satisfying the Gilbert–Varshamov (GV) bound within a random en-
semble of woven codes based on s-partite, s-uniform hypergraphs and constituent block codes was
proved, where s depends only on the code rate. Due to a simple structure of woven graph codes,
such codes can be analyzed with low computational complexity while their minimum distances
are rather close to minimum distances of the best known linear codes of the same lengths and
dimensions. Moreover, there exist linear-time encoding algorithms for this class of codes.
In this paper we generalize the GV bound  to the corresponding bound for woven graph codes
with constituent convolutional codes, namely, the Costello bound. Therefore, we consider woven
graph codes with constituent convolutional codes based on s-partite, s-uniform hypergraphs, in
the following referred to as woven convolutional graph codes. Let the overall constraint length of
woven convolutional graph codes tend towards inﬁnity. Then, within the random ensemble of such
convolutional codes with s ≥ 2, codes satisfying the Costello lower bound on the free distance exist
for any rate. Examples of woven convolutional graph codes with surprisingly large free distances
Supported in part by the Swedish Research Council, Grant no. 621-2007-6281.