Abstract
We give two new constructions of almost difference sets. The first is a generic construction of \((q^{2}(q+1),\,q(q^{2}-1),\,q(q^{2}-q-1),\,q^{2}-1)\) almost difference sets in certain groups of order \(q^{2}(q+1)\) (q is an odd prime power) having (\(\mathbb {F}_{q^{2}},\,+)\) as a subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields \((4p,\,2p+1,\,p,\,p-1)\) almost difference sets in dihedral groups of order 4p where \(p\equiv 3 \ (\mathrm{mod} \ 4)\) is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained produce Cayley graphs which are Ramanujan graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arasu K.T., Ding C., Helleseth T., Kumar P.V., Martinsen H.M.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inf. Theory 47(7), 2934–2943 (2001).
Cai Y., Ding C.: Binary sequences with optimal autocorrelation. Theor. Comput. Sci. 410(24–25), 2316–2322 (2009).
Cao X.: On some expander graphs and algebraic Cayley graphs (2013). arXiv:1306.2690v2.
Cao X., Daying S.: Some nonexistence results on generalized difference sets. Appl. Math. Lett. 21, 797–802 (2008).
Clayton D.: A note on almost difference sets in nonabelian groups. Des. Codes Cryptogr. 72(3), 1–6 (2017).
Cusick T.W., Ding C., Renvall A.: Stream Ciphers and Number Theory. North-Holland Mathematical Library, vol. 55. North-Holland Publishing Co., Amsterdam (1998).
Davis J.A.: Almost difference sets and reversible divisible difference sets. Arch. Math. (Basel) 59(6), 595–602 (1992).
Dillon J.F.: A variations on a scheme of McFarland for noncyclic difference sets. J. Comb. Theory A 40(1), 9–21 (1985).
Ding C.: The differential cryptanalysis and design of natural stream ciphers. In: Fast Software Encryption, pp. 101–115. Springer, Berlin (1994).
Ding C.: Codes from Difference Sets. World Scientific Publishing Co. Pte Ltd., Hackensack (2015).
Ding C., Yuan J.: A family of skew Hadamard difference sets. J. Comb. Theory A 113, 1526–1535 (2006).
Ding C., Helleseth T., Martinsen H.M.: New families of binary sequences with optimal three-level autocorrelation. IEEE Trans. Inf. Theory 47(1), 428–433 (2001).
Ding C., Wang Z., Xiang Q.: Skew Hadamard difference sets from the Ree Tits slice simplectic spreads in \({PG}(3,\,3^{2h+1})\). J. Comb. Theory A 114(5), 867–887 (2007).
Ding C., Pott A., Wang Q.: Constructions of almost difference sets from finite fields. Des. Codes Cryptogr. 72(3), 581–592 (2014).
Ding C., Pott A., Wang Q.: Skew Hadamard difference sets from Dickson polynomials of order 7. J. Comb. Des. 23(10), 436–461 (2014).
Feng T., Xiang Q.: Cyclotomic constructions of skew Hadamard difference sets. J. Comb. Theory A 119, 245–256 (2012).
Feng T., Momihara K., Xiang Q.: Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes. Combinatorica 35(4), 413–434 (2015).
Golomb S.W., Gong G.: Signal Design for Good Correlation. For Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005).
Johnsen E.C.: Skew-Hadamard abelian group difference sets. J. Algebra 1(1), 388–402 (1966).
Jukna S.: Extremal Combinatorics, Texts in Theoretical Computer Science. Springer, Berlin (2011).
Leung K.H., Schmidt B.: Asymptotic nonexistence of difference sets in dihedral groups. J. Comb. Theory A 99(2), 261–280 (2002).
McFarland R.L.: A family of difference sets in non-cyclic groups. J. Comb. Theory A 15, 1–10 (1973).
Metsankyla T.: Catalan’s conjecture: another old Diophantine problem solved. Bull. Am. Math. Soc. 41, 43–57 (2003).
Momihara K.: Inequivalence of skew Hadamard difference sets and triple intersection numbers modulo a prime. Electron. J. Comb. 20(4, P35), 1–19 (2015).
Murty M.R.: Ramanujan graphs. J. Ramanujan Math. Soc. 18(1), 1–20 (2003).
Muzychuk M.: On skew Hadamard difference sets (2010). arXiv:1012.2089v1.
Sidelnikov V.M.: Some \(k\)-valued pseudo-random sequences and nearly equidistant codes. Probl. Pereda. Inf. 5(1), 16–22 (1969).
Weng G., Qiu W., Wang Z., Xiang Q.: Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. Des. Codes Cryptogr. 44, 49–62 (2007).
Zhang Y., Lei J.G., Zhang S.P.: A new family of almost difference sets and some necessary conditions. IEEE Trans. Inf. Theory 52(5), 2052–2061 (2006).
Acknowledgements
The authors would like to thank Professor Qing Xiang for his suggestions which led to some of the work in this manuscript. We would also like to thank the anonymous referees for their valuable comments which helped to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by K. T. Arasu.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors were supported by the National Science Foundation of China under Grant Nos. 11601220 and 61672015.
Rights and permissions
About this article
Cite this article
Michel, J., Wang, Q. Almost difference sets in nonabelian groups. Des. Codes Cryptogr. 87, 1243–1251 (2019). https://doi.org/10.1007/s10623-018-0519-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-018-0519-9