Almost Difference Sets from Unions of Cyclotomic Classes of Order 10

Benedict  M. Estrella,

Almost Difference Sets from Unions of Cyclotomic Classes of Order 10

Abstract:

This study investigated the problem of constructing almost difference sets from single and unions of cyclotomic classes of order 10 (with and without zero) of the finite field GF(q), where q is a prime of the form q = 10n+1 for integer n≥1 and q<1000. Using exhaustive computer searches in Python, the method computed unions of two classes up to nine cyclotomic classes. Moreover, the equivalence of the generated almost difference sets having the same parameters was determined up to complementation. Results showed that a single cyclotomic class formed an almost difference set for q = 31, 71, and 151, while including zero produced an almost difference set for q=11, 31, 71, and 151. Additionally, Paley partial difference sets were constructed from the union of five cyclotomic classes. These findings addressed gaps in the literature on cyclotomy of order 10.

References:

Balmaceda, J. M. P., & Estrella, B. M. (2021). Difference sets from unions of cyclotomic classes of orders 12, 20, and 24. Philippine Journal of Science, 150(6B), 1803–1810. https://doi.org/10.56899/150.6B.16
Davis, J. A. (1992). Almost difference sets and reversible difference sets. Archiv Der Mathematik 59(6), 595–602. https://doi.org/10.1007/BF01194853
Ding, C. (1994). The differential cryptanalysis and design of the natural stream ciphers. In R. Anderson Fast software encryption: Cambridge security workshop, Cambridge, U.K., December 9-11, 1993 proceedings, (pp. 101–115). https://doi.org/10.1007/3-540-58108-1
Ding, C. (2014). Codes from difference sets. World Scientific. https://doi.org/10.1142/9283
Ding, C., Helleseth, T. & Martinsen, H. M. (2001). New families of binary sequences with optimal three-level autocorrelation. IEEE Trans. Information Theory, 47(1), 428–433. https://doi.org/10.1109/18.904555
Ding, C., Pott, A. & Wang, Q. (2014). Constructions of almost difference sets from finite fields. Designs, Codes and Cryptography, 72, 581-592. https://doi.org/10.1007/s10623-012-9789-9
Estrella, B. M. (2022). Construction of difference sets from unions of cyclotomic classes of order N=14. Recoletos Multidisciplinary Research Journal, 10(1), 67-76. https://doi.org/10.32871/rmrj2210.01.04
Nowak, K. (2014, August 30). A survey on almost difference sets. arXiv. https://doi.org/10.48550/arXiv.1409.0114
Nowak, K., Olmez, O., & Song, S. Y. (2013, October 4). Almost difference sets, normally regular digraphs and cyclotomic schemes fom cyclotomy of order twelve. arXiv. https://doi.org/10.48550/arXiv.1310.1164