مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

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 $N^{-1}2^N$ .