Central moments of autocorrelation demerit factors of binary sequences (Daniel Katz, CSUN)

A low autocorrelation binary sequence of length $\ell$ is an $\ell$-tuple of $+1$s and $-1$s that does not strongly resemble any translate of itself.  Such sequences are used in communications and remote sensing for synchronization and ranging, where translation represents time delay.  A single number that indicates how good a sequence is for such purposes, called the merit factor, was introduced by Golay.  Its reciprocal is the demerit factor, which is more natural to analyze due to its connection with norms of polynomials on the complex unit circle.  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. For each positive integer $p$, we derive a formula for the $p$th central moment of the demerit factor for the binary sequences of length $\ell$; this is $\ell^{-2 p}$ times a quasipolynomial function of $\ell$.  The derivations rely on new combinatorial techniques, assisted by group theory and Ehrhart theory, and show that all the central moments are strictly positive for $p\geq 2$ and $\ell \geq 4$. Jedwab’s formula for variance is confirmed, and we go beyond previous results by also deriving an exact formula for the skewness (by hand) and for the kurtosis and the fifth moment (by computer).  We obtain asymptotic values for all central moments in the limit as the length $\ell$ of the sequences tends to infinity.