Synchronization of binary messages

When messages are transmitted as blocks of binary digits, means of locating the beginnings of blocks are provided to keep the receiver in synchronism with the transmitter. Ordinarily, one uses a special synchronizing symbol (which is really a third kind of digit, neither 0 nor 1) for this purpose. The Morse code letter space and the teletype start and stop pulses are examples. If a special synchronizing digit is not available, its function may be served by a short sequence of binary digits
P
which is placed as a prefix to each block. The other digits must then be constrained to keep the sequence
P
from appearing within a block. If blocks of
N
digits (including the prefix
P
) are used, the prefix should be chosen to make large the number
G(N)
of different blocks which satisfy the constraints. Lengthening the prefix decreases the number of "message digits" which remain in the block but also relaxes the constraints. Thus, for each
N
, there corresponds some optimum length of prefix. For each prefix
P
, a generating function, a recurrence formula, and an asymptotic formula for large
N
are found for
G(N)
. Tables of
G(N)
are given for all prefixes of four digits or fewer. Among all prefixes
P
of a given length
A
, the one for which
G(N)
has the most rapid growth is
P = 11 \cdots 1
. However, for this choice of
P
, the table of values of
G(N)
starts with small values;
11 \cdots 1
does not become the best
A
-digit prefix until
N
is very large. At these values of
N
, the
(A + 1) -
digit prefix
11 \cdots 10
is still better. The tables suggest that, for any
N
, a best prefix can always be found in the form
11 \cdots 10
, for suitable
A
. Taking
P = 11 \cdots 10
and
A = [\log_2 (N \log_2 e)]
it is shown that
G(N)
is roughly
0.35N^{-1} X2^N
. This result is near optimal since no choice of
P
can make
G(N)
exceed
N^{-1}2^N
.