The merit factor is an important measure of the collective smallness of the aperiodic autocorrelations of a binary sequence (the other principal measure being the peak sidelobe level). The problem of determining the best value of the merit factor of long binary sequences has resisted decades of attack by mathematicians and communications engineers. In equivalent guise, the determination of the best asymptotic merit factor is an unsolved problem in complex analysis proposed by Littlewood in the 1960s that was studied along largely independent lines for more than twenty years. The same problem is also studied in theoretical physics and theoretical chemistry as a notoriously difficult combinatorial optimisation problem. My 2005 survey paper traces the historical development of the merit factor problem, bringing together results from the various disciplines.

It was established in 1988 that there are infinite families of binary sequences whose asymptotic merit factor attains the value six. Since then no-one has succeeded in finding a set of sequences whose asymptotic merit factor exceeds six. Golay, who coined the term “merit factor”, speculated that even if such a set exists it might never be found, even numerically. But in 2004, Peter Borwein, Stephen Choi and I constructed binary sequences whose merit factor consistently exceeds the value 6.34, for sequence lengths up to several million. Although no-one has yet shown that the merit factor remains above 6.34 when this construction is applied to arbitrarily long sequences, Kai-Uwe Schmidt and I proved in 2010 that a similar construction applied to m-sequences increases the asymptotic merit factor from 3 to greater than 3.34.

M.J.E. Golay (1902-1989) coined the term “merit factor”.

J.E. Littlewood (1885-1977) studied the merit factor as a problem in complex analysis.