The merit factor of a $\{-1, 1\}$ binary sequence measures the collective smallness of its non-trivial aperiodic autocorrelations. Binary sequences with large merit factor are important in digital communications because they allow the efficient separation of signals from noise. It is a longstanding open question whether the maximum merit factor is asymptotically unbounded and, if so, what is its limiting value. Attempts to answer this question over almost sixty years have identified certain classes of binary sequences as particularly important: skew-symmetric sequences, symmetric sequences, and anti-symmetric sequences. Using only elementary methods, we find an exact formula for the mean and variance of the reciprocal merit factor of sequences in each of these classes, and in the class of all binary sequences. This provides a much deeper understanding of the distribution of the merit factor in these four classes than was previously available. A consequence is that, for each of the four classes, the merit factor of a sequence drawn uniformly at random from the class converges in probability to a constant as the sequence length increases.

翻译：$\{-1, 1\}$二元序列的优值因子衡量其非平凡非周期自相关函数的整体小量程度。具有大优值因子的二元序列在数字通信中至关重要，因其能高效实现信号与噪声的分离。一个长期未解的核心问题是：最大优值因子是否渐近无界？若成立，其极限值为何？近六十年来，针对该问题的探索已识别出若干关键二元序列类别：斜对称序列、对称序列与反对称序列。仅通过初等方法，我们推导出这些类别及全体二元序列中互反优值因子均值与方差的精确表达式。这为此前关于四类序列优值因子分布的理解提供了更深刻的洞见。推论表明，对于这四类序列，随着序列长度增加，从该类别均匀随机抽取的序列的优值因子依概率收敛于常数。