Cryptography and Communications

, Volume 10, Issue 2, pp 357–368 | Cite as

Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in C n × Q 8

  • Santiago Barrera AcevedoEmail author
  • Heiko Dietrich


Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q 8 = {±1, ±i, ±j, ±k} was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence between perfect sequences of length n over Q 8q Q 8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in C n × Q 8 with forbidden subgroup C 2; here C m is a cyclic group of order m. We show that if n = p a + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n, 2, 4n, 2n)-relative different set in C n × Q 8 with forbidden subgroup C 2. Lastly, we show that every perfect sequence of length n over Q 8q Q 8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q 8q Q 8).


Perfect sequences Relative difference sets Hadamard matrices Quaternions 

Mathematics Subject Classfication

05B10 05B20 05B30 94C30 


  1. 1.
    Arasu, K.T., de Launey, W.: Two-dimensional perfect quaternary arrays. IEEE Trans. Inf. Theory 47, 1482–1493 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arausu, K.T., de Launey, W., Ma, S.L.: On circular complex hadamard matrices designs. Codes Crypt. 25, 123–142 (2002)CrossRefGoogle Scholar
  3. 3.
    Barrera Acevedo, S., Hall, T.E.: Perfect Sequences of Unbounded Lengths over the Basic Quaternions Lect. Notes. Comput. Sci. SETA2012, pp 159–167 (2012)Google Scholar
  4. 4.
    Davis, J.A., Jedwab, J.: A Unifying Construction of Difference Sets. J. Comb. Theory Ser. A 80, 13–78 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davis, J.A., Jedwab, J., Mowbray, M.: New families of semi-regular relative difference sets. Des. Codes Crypt. 13, 131–146 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Launey, W., Flannery, D.: Algebraic Design Theory Mathematical Surveys and Monographs, 175. AMS, Providence, RI (2011)Google Scholar
  7. 7.
    Horadam, K.J.: Matrices and their applications. Princeton University Press, Princeton (2007)zbMATHGoogle Scholar
  8. 8.
    Ito, N.: On hadamard groups III. Kysushu. J. Math. 1, 369–379 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jungnickel, D.: On automorphism groups of divisible designs. Canad. J. Math. 24, 257–297 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jungnickel, D., Pott, A.: Difference Sets: an introduction. Difference sets, sequences and their correlation properties, pp 259–295. Springer, Netherlands (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kuznetsov, O.: Perfect sequences over the real quaternions. Signal Design and its Applications in Communications 2009. IWSDA ’09. Fourth internat. Workshop 1, 17–20 (2010)Google Scholar
  12. 12.
    Ma, S.L., Ng, W.S.: On Non-existence of Perfect and Nearly Perfect Sequences. International Journal of Information and Coding Theory, 15–38 (2009)Google Scholar
  13. 13.
    Pott, A.: Finite Geometry and Character Theory Lecture Notes in Math, vol. 1601. Springer, Berling-Heidelberg-New York (1995)Google Scholar
  14. 14.
    Pott, A.: A survey on relative difference sets. In: Arasu, K.T., Dillon, J., Harada, K., Sehgal, S., Solomon, R. (eds.) Groups, difference sets, and the monster. Proc. of a Special Research Quarter at Ohio State Uni., Spring 1993, pp 195–232. Walter de Gruyter, Berlin (1996)Google Scholar
  15. 15.
    Rutledge, W.A.: Quaternions and hadamard matrices. Proceedings of the American Mathematical Society 3, 625–630 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schmidt, B.: Williamson matrices and a conjecture of Ito’s. Design, Codes and Cryp. 17, 61–68 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yamada, M.: Hadamard matrices of generalised quaternion type. Discr. Math. 87, 187–196 (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia

Personalised recommendations