Two sequences whose aperiodic autocorrelations sum to zero in all
out-of-phase positions are called a pair of *Golay complementary
sequences*, often defined over an alphabet of size 2 (binary), 4
(quaternary) or 8 (octary).
Golay sequences have been used in many digital communication
applications for which
Barker sequences are not available,
because they are known to exist for infinitely many lengths,
even in the binary case.

In 1999, Jim Davis and I gave an
explicit algebraic normal form
for *m*!/2 · 2^{h(m+1)}
Golay sequences of length 2^{m} over
an alphabet of size 2^{h}.
It was believed for several years that there were no other Golay
sequences for these lengths and alphabets, but in 2005 Ying Li and
Wen Bin Chu unexpectedly found an additional 1024 length 16 quaternary
Golay sequences by computer search.
Frank Fiedler and I showed in 2006 that these new Golay sequences
exist because of a
shared autocorrelation
property of certain quaternary length 8 Golay sequences,
and that they spawn further new quaternary
Golay sequences of length 32, 64, 128, …

Until 2007, the generalisation of a Golay sequence pair to the
multi-dimensional case was studied in detail only by Melanie Dymond.
In 2007, Matthew Parker and I showed that
the projection of a multi-dimensional
Golay array is a Golay sequence, and with
Frank Fiedler we argued in 2008 that an array viewpoint is the
natural way to construct and enumerate Golay sequences and pairs.
This viewpoint accounts for all known Golay sequences and pairs of
length 2^{m},
including all quaternary Golay sequences spawned by
Li and Chu's length 16 examples. Any new sources of sequences, such as
the
length 16 examples over an alphabet of size 6
that I discovered in 2010 with Frank Fiedler and Amy Wiebe,
can be treated simply as additional “seed” pairs in
the multi-dimensional construction.

Construction of two new infinite families of Golay complementary sequences from 6-phase length 16 seed pairs.