The merit factor problem is of practical importance to manifold domains, such as digital communications engineering, radars, system modulation, system testing, information theory, physics, chemistry. However, the merit factor problem is referenced as one of the most difficult optimization problems and it was further conjectured that stochastic search procedures will not yield merit factors higher than 5 for long binary sequences (sequences with lengths greater than 200). Some useful mathematical properties related to the flip operation of the skew-symmetric binary sequences are presented in this work. By exploiting those properties, the memory complexity of state-of-the-art stochastic merit factor optimization algorithms could be reduced from $O(n^2)$ to $O(n)$. As a proof of concept, a lightweight stochastic algorithm was constructed, which can optimize pseudo-randomly generated skew-symmetric binary sequences with long lengths (up to ${10}^5+1$) to skew-symmetric binary sequences with a merit factor greater than 5. An approximation of the required time is also provided. The numerical experiments suggest that the algorithm is universal and could be applied to skew-symmetric binary sequences with arbitrary lengths.

中文翻译：

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关于偏对称二元序列和优值因子问题

品质因数问题对数字通信工程、雷达、系统调制、系统测试、信息论、物理、化学等多个领域具有实际重要性。然而，优值因子问题被认为是最困难的优化问题之一，并且进一步推测随机搜索过程不会为长二进制序列（长度大于 200 的序列）产生高于 5 的优值因子。在这项工作中介绍了一些与倾斜对称二进制序列的翻转操作相关的有用数学特性。通过利用这些特性，最先进的随机优值因子优化算法的内存复杂度可以从 $O(n^2)$ 降低到 $O(n)$。作为概念证明，构建了轻量级随机算法，它可以将伪随机生成的长长度（高达 ${10}^5+1$）的偏对称二进制序列优化为优值因子大于 5 的偏对称二进制序列。所需时间的近似值也是假如。数值实验表明，该算法具有通用性，可应用于任意长度的斜对称二进制序列。