Limiting Moments of Autocorrelation Demerit Factors of Binary Sequences

An aperiodic binary sequence of length $\ell$ is a doubly infinite sequence $f=\ldots,f_{-1},f_0,f_1,\ldots$ with $f_j \in \{-1,1\}$ when $0 \leq j < \ell$ and and $f_j=0$ otherwise. Various problems in engineering and natural science demand binary sequences that do not resemble translates of themselves. The autocorrelation of $f$ at shift $s$ is the dot product of $f$ with the sequence obtained by translating $f$ by $s$ places. The demerit factor of $f$ is the sum of the squares of the autocorrelations at all nonzero shifts for the sequence obtained by normalizing $f$ to unit Euclidean norm. Low demerit factor therefore indicates low self-similarity under translation. We endow the $2^\ell$ binary sequences of length $\ell$ with uniform probability measure and consider the distribution of their demerit factors. Earlier works used combinatorial techniques to find exact formulas for the mean, variance, skewness, and kurtosis of the distribution as a function of $\ell$. These revealed that for $\ell \geq 4$, the $p$th central moment of this distribution is strictly positive for every $p \geq 2$. This article shows that every $p$th central moment is a quasi-polynomial function of $\ell$ with rational coefficients divided by $\ell^{2 p}$. It also shows that, in the limit as $\ell$ tends to infinity, the $p$th standardized moment is the same as that of the standard normal distribution.