ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 3, pp. 198–213.
Pleiades Publishing, Inc., 2008.
Original Russian Text
M.V. Burnashev, H. Yamamoto, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 3, pp. 33–49.
On the Zero-Rate Error Exponent
for a BSC with Noisy Feedback
M. V. Burnashev
and H. Yamamoto
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
The University of Tokyo, Japan
Received March 19, 2008; in ﬁnal form, May 15, 2008
Abstract—For information transmission, a binary symmetric channel is used. There is also
another noisy binary symmetric channel (feedback channel), and the transmitter observes (with-
out delay) all outputs of the forward channel via the feedback channel. Transmission of non-
exponentially many messages is considered (i.e., the transmission rate is zero). The achievable
decoding error exponent for this combination of channels is investigated. 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 similar error exponent of the no-feedback channel.
The described transmission method and the corresponding lower bound for the error exponent
can be improved, as well as extended to positive transmission rates.
1. INTRODUCTION AND MAIN RESULTS
A binary symmetric channel BSC(p) with crossover probability 0 <p<1/2(andq =1− p)is
considered. It is assumed that there is 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 re-transmits all received outputs to
the transmitter). In other words, the feedback channel is “passive.”
Since Shannon’s paper  it has been known that even noiseless feedback does not increase the
capacity of a BSC (or any other memoryless channel). However, feedback can improve the decoding
error probability (or simplify an eﬃcient transmission method). In the case of a BSC with noiseless
feedback, the decoding error probability (or its best error exponent, the channel reliability function)
has been actively studied since Dobrushin , Horstein , and Berlekamp . Also, characteristics
of a number of eﬃcient transmission methods have been investigated (see, for example, [1–10]).
In general, the case of a BSC with noiseless feedback is rather well investigated (though there are
still some important open problems).
The case of noisy feedback was not investigated. It was not even known whether such feedback
can improve the error exponent of the no-feedback case. In this respect, only two recent papers
[11,12] can probably be mentioned, but both of them consider other problems. The paper  uses
variable-length coding (i.e., non-block codes) under another error criterion. Moreover, it is assumed
that at certain moments an error-free feedback mechanism is available. In , a Gaussian channel
with only average power constraint is considered. Such constraint allows using some methods that
are inapplicable in the case of discrete channels.
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00226.