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 pth central moment of the demerit factor for binary sequences of length \(\ell \). These techniques prove that for \(p\ge 2\) and \(\ell \ge 4\), all the central moments are strictly positive. For any given p, one may use the technique to obtain an exact formula for the pth 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.

具有低非周期性自相关的序列用于通信和遥感中的同步和测距。当我们将序列归一化为单位欧几里德长度时，序列的自相关缺点因子是其在每个非零移位处的自相关值的平方幅值之和。 Golay 提出的优点系数是缺点系数的倒数。我们考虑长度为\(\ell \) 的 \(2^\ell \ ) 二元序列的均匀概率测度，并研究这些序列的缺点因子的分布。 Sarwate 和 Jedwab 分别计算了该分布的均值和方差。我们开发了新的组合技术来计算长度为\(\ell \)的二进制序列的缺点因子的第p中心矩。这些技术证明，对于\(p\ge 2\)和\(\ell \ge 4\) ，所有中心矩都是严格正的。对于任何给定的p ，可以使用该技术来获得作为长度函数的缺点因子的p th 中心矩的精确公式。我们的技术通过简短的计算证实了 Jedwab 的方差公式，并且我们还推导出了精确的偏度公式，从而超越了之前的结果。我们的方法的计算机辅助应用还获得了峰度的精确公式（我们在此报告）以及第五中心矩。