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A classical problem of digital sequence design, first studied in the 1950s but still not well understood, is to determine long binary sequences for which the absolute values of the aperiodic autocorrelations are collectively as small as possible. The ideal sequence from this point of view is a Barker sequence, but there is overwhelming evidence that no Barker sequence of length greater than 13 exists. Since a Barker sequence of length greater than 13 must have constant periodic autocorrelations at all non-zero shifts, it is equivalent to a difference set in a cyclic group. The rich structure of the group setting allows the application of techniques and tools from finite field theory, algebraic number theory, character theory, and elsewhere. This has stimulated much research on difference sets, whose study has matured into a fertile body of theory. However the motivating practical problem remains firmly in the aperiodic domain, and has attracted renewed interest in recent years. We survey various responses to the presumed nonexistence of long Barker sequences, namely: multi-dimensional Barker arrays; the peak sidelobe level of binary sequences; the merit factor of binary sequences; Barker sequences over a non-binary alphabet; and pairs of Golay complementary sequences and arrays. 1. The Barker Sequence Conjecture We consider a length s binary sequence to be a one-dimensional matrix A = (A[i]) whose elements satisfy A[i] = { −1 or 1 for 0 ≤ i < s 0 otherwise. The aperiodic autocorrelation function of a length s binary sequence A = (A[i]) is given by

What can be used instead of a Barker sequence?