Normal bases and primitive elements over finite fields

Primitive Normal Values of Rational Functions over Finite Fields

“…Let m = 5, then ρ(2, m) = 1/5 by Lemma 2.5, and from Inequality (8), we get 18,36,72. For these values of n, we choose the values of d and g listed in Table 2 such that the inequality q n/2 > 6W (d) 2 W (g) 2 S holds and get that (2, n) ∈ Q 2 (2, 2) unless n = 9, 18.…”

“…Recently, Liao et al [16] generalized this result to the case where q is any prime power. Kapetanakis [17,18] contributed in this direction by proving the existence of primitive normal pairs α, aα+b cα+d in F q n over F q for q ≥ 23 and n ≥ 17. Anju and R.K. Sharma [19] proved the existence of primitive pairs (α, α 2 + α + 1) in F q n and further showed the existence of primitive pairs (α, α 2 + α + 1) where α is a normal element in F q n over F q .…”

“…For these values of n, we choose the values of d and g listed in Table2such that the inequality q n/2 > 7W (d) 2 W (g) 2 S holds and get that (4, n) ∈ Q 2 (2, 2) unless n = 3, 4, 5, 6, 8, 10, 12, 15.Case k = 3 : In this case m = 1, 3, 7, 9, 21, 63 and Inequality (7) holds for i ≥ 5, when m = 1, for i ≥ 4, when m = 3, for i ≥ 3, when m = 7, 9 and for i ≥ 2, when m = 21, 63. Hence, (8, n) ∈ Q 2 (2, 2) unless n = 3, 4, 6, 7, 8, 9, 12, 14,16,18,21,24,28, 36, 42, 63, 126. For these remaining values, we directly verify the inequality q n/2 > 7W (q n − 1) 2 W (x n − 1) 2 and get that (8, n) ∈ Q 2 (2, 2) unless n…”

An extension of the (strong) primitive normal basis theorem

“…For the remaining cases, namely q ∈ {23, 25} and 17 ≤ m < 67, we compute c q 0 for each pair and check that Eq. (19) is satisfied for all but 16 pairs (q, m). We explicitly check all the remaining pairs and find that only (23, 24) satisfies s = 2.…”

Further Results on the Morgan–Mullen Conjecture

“…Subsequently, Cohen and Huczynska [3] provided a computer-free proof with the help of sieving techniques. Several generalizations of this have also been investigated [2,4,12,13,14].…”