Abstract
In this work, we study cyclic codes that have generators as Fibonacci polynomials over finite fields. We show that these cyclic codes in most cases produce families of maximum distance separable and optimal codes with interesting properties. We explore these relations and present some examples. Also, we present applications of these codes to secret sharing schemes.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40065-017-0171-7/MediaObjects/40065_2017_171_Figa_HTML.gif)
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ashikhmin, A.; Barg, A.; Cohen, G.; Huguet, L.: Variations on minimal codewords in linear codes. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 96–105. Springer, Berlin (1995)
Ashikhmin, A.; Barg, A.: Minimal vectors in linear codes. IEEE Trans. Inform. Theory 44(5), 2010–2017 (1998)
Asmuth, C.; Bloom, J.: A modular approach to key safeguarding. IEEE Trans. Inform. Theory 30(2), 208–210 (1983)
Blakley, G.B.: Safeguarding cryptographic keys. Proc. AFIPS 48, 313–317 (1979)
Burton, D.M.: Elementary Number Theory. Tata McGraw-Hill Education, New York (2006)
Ding, C.; Yuan, J.: Covering and secret sharing with linear codes. Discrete Mathematics and Theoretical Computer Science, pp. 11–25. Springer, Berlin (2003)
Esmaeili, M.; Esmaeili, M.: A Fibonacci-polynomial based coding method with error detection and correction. Comput. Math. Appl. 60(10), 2738–2752 (2010)
Hazewinkel, M. (ed.): Handbook of Algebra, vol. 1. North-Holland, Amsterdam (1995)
Horadam, A.F.: A generalized Fibonacci sequence. Am. Math. Mon. 68(5), 455–459 (1961)
Kautz, W.H.: Fibonacci codes for synchronization control. IEEE Trans. Inform. Theory 11(2), 284–292 (1965)
Lee, G.Y.; Choi, D.H.; Kim, J.S.: Burst-error-correcting block code using Fibonacci code. J. Chungcheong Math. Soc. 22(3), 367–374 (2009)
Li, Z.; Ting, X.; Hong, L.: Secret sharing schemes from binary linear codes. Inform. Sci. 180, 4412–4419 (2010)
Ling, S.; Xing, C.: Coding Theory. A First Course. Cambridge University Press, Cambridge (2004)
MacWilliams, F.J.; Sloane, N.J.A.: The Theory of Error-Correcting Codes, vol. 16. Elsevier, Amsterdam (1977)
Massey, J.L.: Minimal codewords and secret sharing. In: Proceedings of the 6th Joint Swedish–Russian International Workshop on Information Theory (1993)
McEliece, R.J.; Sarwate, D.V.: On sharing secrets and Reed–Solomon codes. Commun. ACM 24(9), 583–584 (1993)
Nyberg, K.: Differentially uniform mappings for cryptography. Advances in Cryptology-EUROCRYPT, vol. 93, pp. 55–64. Springer, New York (1994)
Robinson, D.W.: The Fibonacci matrix modulo \(m\). Fibonacci Q. 1(2), 29–36 (1963)
Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)
Si, W.; Ding, C.: A simple stream cipher with proven properties. Cryptogr. Commun. 4, 79–104 (2012)
Vajda, S.: Fibonacci and Lucas Numbers, and the Golden Section. Ellis Horwood Limited, England (1989)
Wall, D.D.: Fibonacci series modulo M. Am. Math. Mon. 67(6), 525–532 (1960)
Acknowledgements
This research is partially supported by TUBITAK-ARDEB under the project with Grant number 114F388.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Koroglu, M.E., Ozbek, I. & Siap, I. Optimal codes from Fibonacci polynomials and secret sharing schemes. Arab. J. Math. 6, 297–308 (2017). https://doi.org/10.1007/s40065-017-0171-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-017-0171-7