On optimal codes for binary asymmetric channels (Corresp.)

Let
\{ v_{1}, \cdots , v_{N} \}
be a code of
N
vectors whose components consist of
m
ones and
(n - m)
zeros. The vectors are transmitted with equal probability over the asymmetric channel whose error probabilities for transmitted zeros and ones equal respectively
0
and
p, 0 < p < 1
(such channels arise in optical communication). The problem of determining for fixed values of the parameters
N, n, m, p
those codes which achieve minimum error probability is studied. The problem is solved for
N \leq 3
. Partial results are obtained for all
N
.