# Periodic Golay Pairs of Length 72

• Dragomir Ž.-Doković
• Ilias S. Kotsireas
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 133)

## Abstract

We construct supplementary difference sets (SDSs) 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.

## Keywords

Ordinary and periodic Golay pairs Supplementary difference sets Compression method

## Notes

### Acknowledgements

The authors wish to acknowledge generous support by NSERC. This research was enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca). We thank a referee for his suggestions.

## References

1. 1.
Arasu, K.T., Xiang, Q.: On the existence of periodic complementary binary sequences. Des. Codes Crypt. 2, 257–262 (1992)
2. 2.
Borwein, P.B., Ferguson, R.A.: A complete description of Golay pairs for lengths up to 100. Math. Comput. 73(246), 967–985 (2003)
3. 3.
-​Doković, D.Ž.: Cyclic (v; r, s; λ) difference families with two base blocks and v ≤ 50. Ann. Comb. 15, 233–254 (2011)Google Scholar
4. 4.
-​Doković, D.Ž., Kotsireas, I.S.: Compression of periodic complementary sequences and applications. Des. Codes Crypt. 74, 365–377 (2015)Google Scholar
5. 5.
-​Doković, D.Ž., Kotsireas, I.S:. D-optimal matrices of orders 118, 138, 150, 154 and 174 (to appear)Google Scholar
6. 6.
-​Doković, D.Ž., Kotsireas, I.S.: Some new periodic Golay pairs. Numer. Algorithms (to appear). doi: 10.1007/s11075-014-9910-4Google Scholar
7. 7.
Eliahou, S., Kervaire, M., Saffari, B.: A new restriction on the lengths of Golay complementary sequences. J. Comb. Theory Ser. A 55, 49–59 (1990)
8. 8.
Georgiou, S.D., Stylianou, S., Drosou, K., Koukouvinos, C.: Construction of orthogonal and nearly orthogonal designs for computer experiments. Biometrika 101(3), 741–747 (2014)
9. 9.
Koukouvinos, C., Seberry, J.: New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function – a review. J. Stat. Plann. Inference 81, 153–182 (1999)
10. 10.
Turyn, R.J.: Hadamard matrices, Baumert-Hall units, four symbol sequences, puls compression and surface wave encodings. J. Comb. Theory Ser. A 16, 313–333 (1974)

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

1. 1.Department of Pure MathematicsUniversity of WaterlooONCanada
2. 2.Department of Physics & Computer ScienceWilfrid Laurier UniversityONCanada