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 an exact formulas for the kurtosis, which we report here, as well as the fifth central moment.

翻译：具有低非周期自相关性的序列广泛应用于通信和遥感中的同步与测距。序列的自相关劣化因子定义为：将序列归一化为单位欧氏长度后，所有非零移位处自相关值模平方之和。Golay引入的品质因子是劣化因子的倒数。我们考虑长度为$\ell$的$2^\ell$个二进制序列上的均匀概率测度，研究这些序列劣化因子的分布。Sarwate和Jedwab分别计算了该分布的均值与方差。我们发展了新的组合数学技术，用于计算长度为$\ell$的二进制序列劣化因子的$p$阶中心矩。这些技术证明：当$p\geq 2$且$\ell \geq 4$时，所有中心矩严格为正。对于任意给定的$p$，可利用该技术获得劣化因子$p$阶中心矩关于长度$\ell$的精确表达式。我们的方法通过简短计算验证了Jedwab的方差公式，并进一步推导出偏度的精确公式，突破了先前结果。借助计算机辅助应用该方法，我们还得到了峰度（本文报告）以及五阶中心矩的精确公式。