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

On the Skew-Symmetric Binary Sequences and the Merit Factor Problem