Problems of Information Transmission, Vol. 40, No. 3, 2004, pp. 187–194. Translated from Problemy Peredachi Informatsii, No. 3, 2004, pp. 3–12.
Original Russian Text Copyright
2004 by Zyablov, Johannesson, Pavlushkov.
Detecting and Correcting Capabilities
of Convolutional Codes
V. V. Zyablov
, R. Johannesson
, and V. A. Pavlushkov
Institute for Information Transmission Problems, RAS, Moscow
Department of Information Technology, Lund University, Sweden
Received December 2, 2003; in ﬁnal form, May 11, 2004
Abstract—A convolutional code can be used to detect or correct inﬁnite sequences of errors
or to correct inﬁnite sequences of erasures. First, erasure correction is shown to be related to
error detection, as well as error detection to error correction. Next, the active burst distance is
exploited, and various bounds on erasure correction, error detection, and error correction are
obtained for convolutional codes. These bounds are illustrated by examples.
In this paper we discuss error-detecting, error-correcting, and erasure-correcting capabilities of
convolutional codes. Since code sequences of convolutional codes are of inﬁnite length, we consider
error and erasure sequences that are also of inﬁnite length. When we investigate these detecting
and correcting capabilities, we assume that the code sequences are periodically terminated during
communication, and that the period is chosen to be suﬃciently long so that we can neglect the
eﬀects of rate reduction. This assumption is normally fulﬁlled in practice. It is quite natural to use
active distances  as a tool to obtain lower bounds on maximal detectable or correctable densities
of errors and erasures. This approach allows us to focus on distance properties of convolutional
codes in contrast to the more general view in Forney’s paper .
In Section 2, we consider arbitrary error and erasure sequences and connect the error-detecting
capability to the erasure-correcting capability for an arbitrary linear code. We also give rather
general relations between the error-correcting and error-detecting capability. In Section 3, we
exploit the active burst distance and obtain lower bounds on detecting and correcting capabilities
for sparse sequences of errors and erasures respectively. In Section 4, these results are extended to
the situation where we have sequences of error bursts or erasure bursts.
2. DETECTION AND CORRECTION OF ARBITRARY SEQUENCES
OF ERRORS AND ERASURES
In this section we show how erasure correction and error correction are related to error detection.
First we introduce our notations. Although the results can be formulated for a nonbinary alphabet,
we for simplicity restrict the presentation to the binary case.
Consider a code C and a set of error sequences S such that, for all codewords c
, and for all error sequences e
. Clearly, any error
Supported in part by the Royal Swedish Academy of Sciences in cooperation with the Russian Academy
of Sciences; the Swedish Research Council, Grant no. 2003-3262; and the Graduate School in Personal
Computing and Communication PCC++.
2004 MAIK “Nauka/Interperiodica”