关于反对称二元序列与价值因子问题
On the Skew-Symmetric Binary Sequences and the Merit Factor Problem
摘要：
品质因子问题在数字通信工程、雷达、系统调制、系统测试、信息论、物理、化学等领域都具有重要的实际意义。然而，品质因数问题被称为最困难的优化问题之一，并且进一步证明随机搜索过程对于长二进制序列（长度大于200的序列）不会产生高于5的品质因数。本文给出了与反对称二元序列翻转运算有关的一些有用的数学性质。通过利用这些性质，最先进的随机品质因子优化算法的存储器复杂度可以从$O(n^2)$降低到$O(n)$。作为概念证明，构造了一种轻量级随机算法，该算法可以将伪随机生成的具有长长度（高达${10}^5+1$）的反对称二进制序列优化为优值因子大于5的反对称二进制序列。还提供了所需时间的近似值。数值实验表明，该算法具有通用性，可应用于任意长度的反对称二元序列。
英文摘要：
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.
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