Generating good synchronization sequences
A synchronization sequence generally needs the property that its autocorrelation function resembles an impulse.  There are two possible autocorrelation functions that can be considered. For a (real-valued) sequence $x$ of length $N$, the periodic autocorrelation function is
$$R_x[n] = \sum_{k=0}^{N-1}x[k]x[k+n]$$
where the sequence is assumed to extend periodically so that 
$x[k+n] = x[k+n\bmod N]$.  In this case, one commonly used solution to
choose $x$ to be a maximal-length linear binary feedback shift-register
(LFSR) sequence of
period $N = 2^m-1$ whose periodic autocorrelation function is
$$R_x[n] = \begin{cases} N, &amp; n \equiv 0 \bmod N,\\ -1, &amp; n \not\equiv 0 \bmod N,
\end{cases}$$
which resembles an impulse train of period $N$ except that the "out-of-phase"
value of the periodic autocorrelation function is not identically
zero, but is small (and constant) 
compared to the "in-phase" value of $N$.

The other form of autocorrelation function is the aperiodic autocorrelation 
function which is given by
$$C_x[n] = \begin{cases}\sum_{k=0}^{N-1-n}x[k]x[k+n], &amp; 0 \leq n &lt; N,\\
\sum_{k=0}^{N-1+n}x[k-n]x[k], &amp; -N &lt; n &lt; 0,\\
0, &amp; |n| \geq N.
\end{cases}$$
Now, for binary sequences where $x$ takes on values in $\{+1,-1\}$,
the value of $C_x[n]$ alternates between even and odd integers as
$n$ varies from $-(N-1)$ to $(N-1)$, and so a "flat" out-of-phase
aperiodic autocorrelation function is impossible to achieve. 
The closest we can hope to come is to find sequences for which
$|C_x[n]| \leq 1$ for $1\leq n &lt; N$.  Sequences with this property
are called Barker sequences, and the longest known Barker
sequence is of length $13$.  it is widely believed, though not
proven, that Barker sequences of longer lengths do not exist,
and numerical studies show that if such a sequence exists, it
it must be so long that it will not be be usable in any practical
scheme as a synchronization preamble.