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. Previous researchers have 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 $\ell$. The previously obtained formula for variance is confirmed by our technique with a short calculation, and we demonstrate that our techniques go beyond this by also deriving an exact formula for the skewness.

翻译：低非周期自相关的序列用于通信和遥感中的同步与测距。序列的自相关劣化因子是指将其归一化为单位欧氏长度后，所有非零移位处自相关值平方模的和。由Golay引入的优值因子是劣化因子的倒数。我们考虑长度为$\ell$的$2^\ell$个二元序列上的均匀概率测度，并研究这些序列劣化因子的分布。先前的研究人员已计算了该分布的均值和方差。我们开发了新的组合技术，用于计算长度为$\ell$的二元序列劣化因子的第$p$阶中心矩。这些技术证明，对于$p \geq 2$且$\ell \geq 4$，所有中心矩均为严格正值。对于任意给定的$p$，可利用该技术获得劣化因子第$p$阶中心矩关于长度$\ell$的精确表达式。我们通过简短计算确认了先前得到的方差公式，并进一步推导出偏度的精确表达式，从而展示了这些技术的扩展能力。