ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 1, pp. 1–4.
Pleiades Publishing, Inc., 2009.
Original Russian Text
V.M. Blinovsky, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 1, pp. 3–7.
Plotkin Bound Generalization
to the Case of Multiple Packings
V. M. Blinovsky
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received September 30, 2008; in ﬁnal form, November 13, 2008
Abstract—We obtain an upper bound on the cardinality of a multiple packing in the case of
a large decoding radius.
be the Hamming space of binary sequences of length n with metric d
x, y ∈ F
. A sphere of radius t centered at y ∈ F
is the set B
is an arbitrary set C⊂F
.AnL-packing with spheres of radius t (or an (L, t)-paching) is a code
C = C(t, L) such that for any vector y ∈ F
|C ∩ B
It is easily seen that an (L, t)-packing C(t, L) is also characterized by the property that for any
betherateofthecodeC(L, t), and let τ = t/n. In  there are obtained
upper bounds on
R(τ,L) = lim sup
which can be written in a parametric form as follows:
R(τ,L) ≤ 1 − H(λ), (1)
where λ ∈ [0, 1/2] is found from the equation
2i − 2
i − 1
(λ(1 − λ))
Furthermore, in  (see also ) there are obtained lower bounds on R(τ, L) and it is shown that,
for the zero rate (the size of the code still tending to inﬁnity, but nonexponentially), the bound
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00226.