  Sequences with low aperiodic autocorrelation are used in communications and
remote sensing for synchronization and ranging. The autocorrelation demerit
factor of a sequence is the sum of the squared magnitudes of its
autocorrelation values at every nonzero shift when we normalize the sequence to
have unit Euclidean length. The merit factor, introduced by Golay, is the
reciprocal of the demerit factor. We consider the uniform probability measure
on the $2^\ell$ binary sequences of length $\ell$ and investigate the
distribution of the demerit factors of these sequences. Sarwate and Jedwab have
respectively calculated the mean and variance of this distribution. We develop
new combinatorial techniques to calculate the $p$th central moment of the
demerit factor for binary sequences of length $\ell$. These techniques prove
that for $p\geq 2$ and $\ell \geq 4$, all the central moments are strictly
positive. For any given $p$, one may use the technique to obtain an exact
formula for the $p$th central moment of the demerit factor as a function of the
length $\ell$. Jedwab's formula for variance is confirmed by our technique with
a short calculation, and we go beyond previous results by also deriving an
exact formula for the skewness. A computer-assisted application of our method
also obtains exact formulas for the kurtosis, which we report here, as well as
the fifth central moment.
