Linear Difference Equations and Periodic Sequences over Finite Fields
- 79 Downloads
First, we study linear equations over finite fields in general. An explicit formula for a common period is found for every solution of a linear difference equation over a finite field. It will help to estimate the p-adic modulus of polynomial roots. Second, we focus our attention on periodic sequences over finite fields and Hamiltonian cycles in de Bruijn directed graph.
KeywordsJordan multiplicative decomposition Characteristic equations Lucas’ congruence Minimal polynomial Trace representation Max-sequences
Mathematics Subject Classification (2010)12E20 12Y05
The author would like to express his sincere thank to the referee for reading carefully the manuscript and providing some suggestions that have been implemented in the final version of the paper. Deepest appreciation is extended towards the NAFOSTED (the National Foundation for Science and Technology Development in Vietnam) for the financial support.
- 5.Goodman, R., Wallach, N.R.: Representations and invariants of the classical groups. Cambridge University Press (2001)Google Scholar
- 6.Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Pure Appl. Math., Vol. 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1978)Google Scholar
- 7.Kocic, V., Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications. Mathematics and its Applications, Vol. 256. Kluwer Academic Publishers Group, Dordrecht (1993)Google Scholar
- 9.Lidl, R., Niederreiter, H.: Finite fields. With a foreword by P. M. Cohn. Second edition, Vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar
- 10.Niederreiter, H.: Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. vi+241 pp. ISBN: 0-89871-295-5Google Scholar
- 12.Robert, A.M.: A Course in p−adic analysis. Springer (2000)Google Scholar