On synchronizable and PSK-synchronizable block codes

A (finite) code is called synchronizable of
M
th order if, and only if, there exists a least positive integer
M
such that the knowledge of the last
M
letters of any message suffices to determine a separation of code words. This condition is less constraining than the comma-free restriction. Sections I-IV is a description of a way of selecting a synchronizable block code
C_{\sigma , n}
for a given alphabet with
\delta
letters and a given code-word length n. It is proved that
C_{\sigma , n}
is maximal in the sense that it contains as many words as possible; and an expression is given for the number of words in
C_{\sigma , n}
.
^{1}
Sections V-VIII is a description of a construction of phase-shift-keying synchronizable codes. In the PSK case the letters themselves cannot be identified by the receiver. Only the differences between successive letters can be detected. Thus, if the sequence
X_{1},X_{2}, \cdots , X_{l}
is transmitted, then the observed sequence in the receiver is
X_{2} - X_{1}, X_{3}- X_{2}, \cdots , X_{i+1}- X_{i}, \cdots , X_{l} - X_{l-1}
. The letters are assumed to be the integers mod
\delta
; and
X_{i+1} - X_{i}
is taken mod
\delta
. This situation occurs in the reading of phase-modulated signals. The construction of these codes (called
K_{\sigma , n}
) is based on
C_{\sigma , n}
.