  We construct supplementary difference sets (SDS) with parameters
$(72;36,30;30)$. These SDSs give periodic Golay pairs of length 72. No periodic
Golay pair of length 72 was known previously. The smallest undecided order for
periodic Golay pairs is now 90. The periodic Golay pairs constructed here are
the first examples having length divisible by a prime congruent to 3 modulo 4.
The main tool employed is a recently introduced compression method. We observe
that Turyn's multiplication of Golay pairs can be also used to multiply a Golay
pair and a periodic Golay pair.
