Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T11:29:11.075Z Has data issue: false hasContentIssue false
  • English
  • Français

QUALIFIED DIFFERENCE SETS FROM UNIONS OF CYCLOTOMIC CLASSES

Part of: Designs and configurations Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  17 April 2009

KEVIN BYARD  and
KEVIN BROUGHAN
KEVIN BYARD*
Affiliation:
Institute of Information and Mathematical Sciences, Massey University, Albany, North Shore, Auckland, New Zealand (email: k.byard@massey.ac.nz)
KEVIN BROUGHAN
Affiliation:
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand (email: kab@waikato.ac.nz)
*
✉For correspondence; e-mail: k.byard@massey.ac.nz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Qualified difference sets (QDS) composed of unions of cyclotomic classes are discussed. An exhaustive computer search for such QDS and modified QDS that also possess the zero residue has been conducted for all powers n=4,6,8 and 10. Two new families were discovered in the case n=8 and some new isolated systems were discovered for n=6 and n=10.

Keywords

qualified residue difference setsdifference setscyclotomy

MSC classification

Secondary: 05B10: Difference sets (number-theoretic, group-theoretic, etc.) 11T22: Cyclotomy
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 147 - 158
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Bateman, P. T. and Horn, R. A., ‘A heuristic asymptotic formula concerning the distribution of prime numbers’, Math. Comp. 16 (1962), 363–367.Google Scholar
[2] Baumert, L. D., Cyclic Difference Sets, Lecture Notes in Mathematics, 182 (Springer, New York, 1971).Google Scholar
[3] Berndt, B. C., Evans, R. J. and Williams, K. S., Gauss and Jacobi Sums, Canad. Math. Soc. Ser. Monogr. Adv. Texts series, 21 (Wiley, New York, Toronto, 1998).Google Scholar
[4] Byard, K., ‘Synthesis of binary arrays with perfect correlation properties – coded aperture imaging’, Nucl. Instrum. Methods Phys. Res. A 336 (1993), 262–268.Google Scholar
[5] Caroli, E., Stephen, J. B., Di Cocco, G., Natalucci, L. and Spizzichino, A., ‘Coded aperture imaging in x- and gamma-ray astronomy’, Space Sci. Rev. 45 (1987), 349–403.CrossRefGoogle Scholar
[6] Dickson, L. E., ‘Cyclotomy, higher congruences and Waring’s problem’, Amer. J. Math. 57 (1935), 391–424.CrossRefGoogle Scholar
[7] Hall Jr, M., ‘A survey of difference sets’, Proc. Amer. Math. Soc. 7 (1956), 975–986.CrossRefGoogle Scholar
[8] Hardy, G. H. and Littlewood, J. E., ‘Some problems of ‘Partitio numerorum’; III: on the expression of a number as a sum of primes’, Acta Math. 44 (1923), 1–70.Google Scholar
[9] Hayashi, H. S., ‘Computer investigation of difference sets’, Math. Comp. 19 (1965), 73–78.CrossRefGoogle Scholar
[10] Jennings, D. and Byard, K., ‘An extension for residue difference sets’, Discrete Math. 167/168 (1997), 405–410.Google Scholar
[11] Jennings, D. and Byard, K., ‘Qualified residue difference sets with zero’, Discrete Math. 181 (1998), 283–288.Google Scholar
[12] Klemperer, W. K., ‘Very large array configurations for the observation of rapidly varying sources’, Astron. Astrophys. Suppl. 15 (1974), 449–451.Google Scholar
[13] Lehmer, E., ‘On residue difference sets’, Canad. J. Math. 5 (1953), 425–432.CrossRefGoogle Scholar
[14] Lehmer, E., ‘On the number of solutions of u k+D≡w 2(modp)’, Pacific J. Math. 5 (1955), 103–118.CrossRefGoogle Scholar
[15] Luke, H. D., Bomer, L. and Antweiler, M., ‘Perfect binary arrays’, Signal Process. 17 (1989), 69–80.Google Scholar
[16] Mollin, R. A., Algebraic Number Theory (Chapman and Hall/CRC, London, 1999).Google Scholar
[17] Rogers, W. L., Koral, K. F., Mayans, R., Leonard, P. F., Thrall, J. H., Brady, T. J. and Keyes, J. W., ‘Coded aperture imaging of the heart’, J. Nucl. Med. 21 (1980), 371–378.Google ScholarPubMed