Periodic Golay pairs of length 72

TL;DR

This paper presents the first known periodic Golay pairs of length 72, constructed using a new compression method, expanding the understanding of Golay pairs in lengths divisible by primes congruent to 3 modulo 4.

Contribution

The authors construct the first periodic Golay pairs of length 72 using a novel compression technique, advancing the field of combinatorial design theory.

Findings

First known periodic Golay pairs of length 72.

Use of a new compression method for construction.

Extension of Turyn's multiplication to periodic Golay pairs.

Abstract

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.