ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 2, pp. 103–121.
Pleiades Publishing, Inc., 2010.
Original Russian Text
M.V. Burnashev, H. Yamamoto, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 2, pp. 3–23.
On the Reliability Function for a BSC
with Noisy Feedback
M. V. Burnashev
and H. Yamamoto
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
University of Tokyo, Japan
Received August 24, 2009; in ﬁnal form, January 21, 2010
Abstract—A binary symmetric channel is used for information transmission. There is also an-
other noisy binary symmetric channel (feedback channel), and the transmitter observes without
delay all outputs of the forward channel via the feedback channel. Transmission of an exponen-
tial number of messages is considered (i.e., the transmission rate is positive). The achievable
decoding error exponent for this combination of channels is studied. It is shown that if the
crossover probability of the feedback channel is less than a certain positive value, then the
achievable error exponent is better than the decoding error exponent of a channel without
1. INTRODUCTION AND MAIN RESULTS
The binary symmetric channel BSC(p) with crossover probability 0 <p<1/2(andq =1− p)
is considered. It is assumed that there is also a feedback BSC(p
) channel, and the transmitter
observes (without delay) all outputs of the forward BSC(p) channel via the noisy feedback channel.
No coding is used in the feedback channel (i.e., the receiver simply resends all received outputs to
the transmitter). In other words, the feedback channel is “passive” (see Fig. 1).
We consider the case where the overall transmission time n and M = e
} are given. After time n, the receiver makes a decision
θ on a transmitted message.
We are interested in the best possible decoding error exponent (and whether it can exceed a similar
exponent of a channel without feedback).
Such model was considered in , where the case of a nonexponential (in n)numberM (i.e.,
R = 0) was investigated. In the present paper we consider the case of M = e
, R>0, strength-
ening the methods of . The main diﬀerence is that now M is exponential in n, and therefore
we need much more accurate investigation of the decoding error probability. Moreover, if M is
nonexponential in n, then we know the best code for use during phase I; namely, this is an “almost
equidistant” code (i.e., all its codeword distances are n/2+o(n)). If R>0, then we do not know
the best code and therefore choose a code randomly.
Some results for channels with noiseless feedback can be found in [2–12]; in the noisy feedback
case, in [13, 14] (see also a discussion in ).
We show that if the crossover probability p
of the feedback channel BSC(p
certain positive value p
(p, R), then it is possible to improve the best error exponent E(R, p)of
Supported in part by the Russian Foundation for Basic Research, project nos. 06-01-00226 and 09-01-