Binary sequences whose aperiodic autocorrelations are collectively small have been sought since the 1950s as modulating signals in digital communications. Two popular measures of smallness are the merit factor and the peak sidelobe level. Studies of these two measures, often motivated by numerical experiments, involve diverse mathematical tools including Fourier analysis, estimation of character sums, estimation of the number of lattice points in polyhedra, finite fields, and probabilistic methods. Despite a large and growing body of theory, numerical experiments suggest that there are unexplained infinite families of binary sequences having record asymptotic merit factor or peak sidelobe level.

I shall describe the merit factor problem and peak sidelobe level problem from scratch, outline their history, and explain the intriguing numerical experiments whose results await explanation.