Abstract
Let n be a positive integer. A nonzero element γ of the finite field F of order q = 2n is said to be “strongly primitive” if every element (aγ + b)/(cγ + d), with a,b,c,d in {0,1} and ad − bc not zero, is primitive in the usual sense. We show that the number N of such strongly primitive elements is asymptotic to θθ′·q where θ is the product of (1 − 1/p) over all primes p dividing (q − 1) and θ′ is the product of (1 − 2/p) over the same set.
Using this result and the accompanying error estimates, with some computer assistance for small n, we deduce the existence of such strongly primitive elements for all n except n = 1, 4, 6. This extends earlier work on Golomb’s conjecture concerning the simultaneous primitivity of γ and γ + 1.
We also discuss analogous questions concerning strong primitivity for other finite fields.
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References
Biggs, N.: Algebraic graph theory, 2nd edn. Cambridge Mathematical Lectures. Cambridge University Press, Cambridge (1993)
Carlitz, L.: Sets of primitive roots. Compositio Math. 13, 65–70 (1956)
Cohen, S.D., Mullen, G.L.: Primitive elements in finite fields and Costas arrays. Appl. Algebra Engrg. Comm. Comput. 2(1), 45–53 (1991)
Golomb, S.W.: Algebraic constructions for Costas arrays. J. Combin. Theory Ser. A 37(1), 13–21 (1984)
Lidl, R., Niederreiter, H.: Finite fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)
Moreno, O.: On the existence of a primitive quadratic of trace 1 over GF(p m). J. Combin. Theory Ser. A 51(1), 104–110 (1989)
Moreno, O., Sotero, J.: Computational approach to conjecture A of Golomb. In: Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989), vol. 70, pp. 7–16 (1990)
Tutte, W.T.: Selected papers of W. T. Tutte, vol. I. McCarthy, D., Stanton, R.G. (eds.). Charles Babbage Research Center, Winnipeg (1979)
Weil, A.: On some exponential sums. Proc. Nat. Acad. Sci. 34, 204–207 (1948)
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Goldstein, D., Hales, A.W. (2007). Strongly Primitive Elements. In: Golomb, S.W., Gong, G., Helleseth, T., Song, HY. (eds) Sequences, Subsequences, and Consequences. Lecture Notes in Computer Science, vol 4893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77404-4_3
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DOI: https://doi.org/10.1007/978-3-540-77404-4_3
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