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.

翻译：在通信和遥感领域，低非周期自相关序列被用于同步与测距。序列的自相关劣质因子定义为：当序列经归一化处理使其具有单位欧几里得长度后，其在所有非零偏移处的自相关值平方幅度之和。由 Golay 引入的优质因子即为劣质因子的倒数。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，并研究这些序列劣质因子的分布。Sarwate 与 Jedwab 分别计算了该分布的均值与方差。我们发展了新的组合技术以计算长度为 $\ell$ 的二进制序列劣质因子的 $p$ 阶中心矩。这些技术证明，对于 $p\geq 2$ 且 $\ell \geq 4$ 的情形，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质因子 $p$ 阶中心矩关于长度 $\ell$ 的精确公式。我们通过简短计算验证了 Jedwab 的方差公式，并进一步推导了偏度的精确公式，超越了先前的研究结果。通过计算机辅助应用本方法，我们还得到了峰度以及五阶中心矩的精确公式，并在此予以报告。