Abstract
We determine the trace function representation of Hall's sextic residue sequences of period p ≡ 7 (mod 8). Current status of a conjecture regarding the existence of Hadamard sequences is briefly discussed.
This work was supported in part by BK21 Korea.
This work was performed while J.-H. Kim was with Department of Electrical and Electronics Engineering, Yonsei University, Seoul, Korea.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. D. Baumert, Cyclic Difference Sets, Lecture notes in mathematics, vol. 182, Springer-Verlag, New York, 1971.
C. Ding, “Linear complexity of generalized cyclotomic binary sequences of order 2,” Finite Fields and Their Applications, vol. 3, no. 2, pp. 159–174, 1997.
C. Ding, T. Helleseth, and W. Shan, “On the Linear Complexity of Legendre Sequences,” IEEE Transactions on Information Theory vol. 44, no. 3, pp. 1276–1278, 1998.
S. W. Golomb, Shift Register Sequences, Aegean Park Press, Laguna Hills, CA, 1982.
S. W. Golomb, “Construction of Signals with Favourable Correlation Properties,” in Survey in Combinatorics, A. D. Keedwell, Editor; LMS Lecture Note Series 166, Cambridge University Press, pp. 1–40, 1991.
S. W. Golomb and H. -Y. Song, “A conjecture on the existence of cyclic Hadamard difference sets,” Journal of statistical planning and inference, vol. 62, pp. 39–41, 1997.
G. Gong, Lecture Notes on Sequence Design and Analysis, pre-print, http://calliope.uwaterloo.ca/~ggong/~ggong
M. Hall Jr., “A Survey of Difference Sets,” Proc. Amer. Math. Soc., vol. 7, pp. 975–986, 1956.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, second edition, 1990.
Dieter Jungnickel, “Difference Sets,” in Contemporary Design Theory edited by J. H. Dinitz and D. R. Stinson, pp. 241–324, John Wiley & Sons, Inc., New York, 1992.
J. -H. Kim and H. -Y. Song, “Existence of Cyclic Hadamard Difference Sets and its Relation to Binary Sequences with Ideal Autocorrelation,” Journal of Communications and Networks, vol. 1, no. l, pp. 14–18, March 1999.
J. -H. Kim and H. -Y. Song, “On the linear complexity of hall’s sextic residue sequences,” IEEE Transaction on Information Theory, vol. 47, no. 5, pp. 2094–2096, June 2001.
J. -H. Kim and H. -Y. Song, “Trace Representation of Legendre Sequences,” Designs, Codes and Cryptography, vol. 24, no. 3, pp. 343–348, December 2001.
H. -K. Lee, J. -S. No, H. Chung, K. Yang, J. -H. Kim, and H. -Y. Song, “Trace function representation of Hall’s sextic residue sequences and some new sequences with ideal autocorrelation,” in Proceedings of APCC′97. APCC, Dec. 1997, pp. 536–540.
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, vol. 20, Addison-Wesley, Reading, MA, 1983.
J. -S. No, H. -K. Lee, H. Chung, H. -Y Song and K. Yang, “Trace Representation of Legendre Sequences of Mersenne Prime Period,” IEEE Transactions on Information Theory, vol. 42, no. 6, pp. 2254–2255, Nov. 1996.
M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications Handbook, revised edition, McGraw-Hill, 1994.
H. -Y. Song and S. W. Golomb, “On the existence of cyclic Hadamard difference sets,” IEEE Trans. Inform. Theory, vol. 40, no. 4, pp. 1266–1268, July 1994.
R. G. Stanton and D. A. Sprott, “A Family of Difference Sets,” Canadian Journal of Mathematics, vol. 10, pp. 73–77, 1958.
R. Turyn, “The linear generation of the Legendre sequences,” Journal of Soc. Ind. Appl. Math., vol. 12, no. 1, pp. 115–117, 1964.
Z. Dai, G. Gong, and H.-Y. Song, “Trace representation of binary Jacobi sequences” preprint, 2002.
Z. Dai, G. Gong, and H.-Y. Song, “Trace representation and linear complexity of binary e-th residue sequences of period p,” preprint, 2002.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kim, JH., Song, HY., Gong, G. (2003). Trace Representation of Hall’s Sextic Residue Sequences of Period P ≡ 7 (mod 8). In: No, JS., Song, HY., Helleseth, T., Kumar, P.V. (eds) Mathematical Properties of Sequences and Other Combinatorial Structures. The Springer International Series in Engineering and Computer Science, vol 726. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0304-0_4
Download citation
DOI: https://doi.org/10.1007/978-1-4615-0304-0_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5013-2
Online ISBN: 978-1-4615-0304-0
eBook Packages: Springer Book Archive