In communications engineering, an ideal binary sequence is a *Barker
sequence*, for which all the out-of-phase aperiodic autocorrelations
have magnitude at most 1.
Although Barker sequences can easily be found for lengths 2, 3, 4,
5, 7, 11 and 13, there is overwhelming evidence that longer Barker
sequences do not exist.

Given both the potential usefulness and the scarcity of Barker sequences,
one practical option is to relax the Barker condition by seeking binary
sequences whose aperiodic autocorrelations are collectively small,
as measured by the
merit factor
or the
peak sidelobe level.
A second option is to seek two sequences whose aperiodic
autocorrelations sum to zero in all out-of-phase positions,
known as a pair of
Golay complementary sequences.
A third option is to generalise the Barker definition to
multi-dimensional binary arrays.
This was proposed by Alquaddoomi and Scholtz for two-dimensional arrays
in 1989, but after theoretical analysis and computational search they
conjectured that an
*s×t* Barker array with *s*, *t* > 1 exists only
when *s=t*=2.

In 1993, I found combinatorial restrictions on the possible sizes of an
*s×t* Barker array when *st* is
even
and, with Sheelagh Lloyd and
Miranda Mowbray, when *st* is
odd.
In 2006 Jim Davis, Ken Smith and I revisited the problem and found a
complete
proof of the two-dimensional Barker array conjecture
using only elementary methods.

Soon afterwards, Matthew Parker and I found a short proof that the two-dimensional result implies there are no Barker arrays having more than two dimensions, as conjectured by Dymond in 1992.

In 2015, Jason Bell, Mahdad Khatirinejad, Kai-Uwe Schmidt and I combined combinatorial arguments and algebraic number theory to establish severe restrictions on the possible sizes of a 3-phase Barker array.

A length 13 Barker sequence.

A 2×2 Barker array.