ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 2, pp. 103–113.
Pleiades Publishing, Inc., 2016.
Original Russian Text
V.S. Lebedev, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 2, pp. 3–14.
Coding with Noiseless Feedback
V. S. Lebedev
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
Received March 30, 2015; in ﬁnal form, March 3, 2016
Abstract—We consider the problem of error correction using nonbinary codes and assuming
noiseless feedback. This is equivalent to the searching with lies problem. We improve the
algorithm proposed by Ahlswede, Deppe, and Lebedev in .
The searching with lies problem in its classical setting can be formulated as follows. How many
yes/no questions one has to ask to determine a secret number in the range from 1 to M if among
the answers there are at most t false ones? The searching can be either adaptive or nonadaptive.
If all the questions are asked at once and then we get all answers at once, such testing is said to
be nonadaptive. If we may use results of preceding tests when asking a new question, the testing
is said to be adaptive. In the present paper we are interested in adaptive testing only.
This problem is called the Ulam problem, or the R´enyi–Ulam problem, and it has many impor-
tant applications (see ). The guessing problem with possible false answers was ﬁrst formulated
by Hungarian mathematician Alfred R´enyi . An important role in studying this problem was
played by results obtained by Berlekamp . The problem became popular after American mathe-
matician Stanislaw Ulam asked this question for M =10
in his autobiography . For this value
of M (more precisely, for M =2
), the problem has been solved. The minimum number N (2
of questions in the adaptive testing problem is given in the table
t 0 1 2 3 4 5 6 7 8 9
,t) 20 25 29 33 37 40 43 46 50 53 ,
and for t ≥ 8wehaveN(2
,t)=3t + 26.
For an arbitrary value of M, a precise answer is known for small values of t only (see [6, 7]). An
asymptotically tight answer was obtained in .
We are interested in generalizing this problem to the q-ary case (see ). Consider a set of M
elements. We divide the set [M]intoq subsets S
. An answer shows to which group
a secret number belongs. How many questions one has to ask to determine a secret number in the
range from 1 to M if among the answers there can be at most t false ones?
In  a transmission algorithm was proposed which extends the results of Berlekamp to the q-ary
case. No asymptotically tight answer for an arbitrary value of t has been found. It is interesting that
The research was carried out at the Institute for Information Transmission Problems of the Russian
Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.