Given a finite field Fq, positive integer n and an Fq-affine space A⊆Fqn, we show that result of Katz provides new bound on the sum ∑a∈Aχ(a), where χ is multiplicative character Fqn. The paper focused applicability this estimate to results regarding existence special primitive elements in In particular, obtain substantial improvements previous works.

Let Fv and Fe be sets of faulty vertices and faulty edges, respectively, in the folded hypercube FQn so that |Fv| + |Fe| ≤ n − 2, for n ≥ 2. Choose any fault-free edge e. If n ≥ 3 then there is a fault-free cycle of length l in FQn containing e, for every even l ranging from 4 to 2 − 2|Fv |; if n ≥ 2 is even then there is a fault-free cycle of length l in FQn containing e, for every odd l rangi...

Let n≥2 be an integer and let Fq the finite field with q elements, where is a prime power. Given Fq-affine hyperplanes A1,…,An of Fqn in general position, we study existence distribution primitive elements Fqn, avoiding each Ai. We obtain both asymptotic concrete results, relating to past works on digits over fields.

Diagnosability plays a crucial role in measuring the reliability and fault tolerance of interconnection networks. The g-extra conditional diagnosability of multiprocessor systems is a new diagnosability, which is more accurate than the classical diagnosability. In this paper, we show that the g-extra conditional diagnosability t̃g(FQn) of the folded hypercube FQn is (g + 1)n − ( g 2 ) + 1 under ...

and infinitely many integers n, multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglogn), division with O(n log n loglogn) operations in Fq , and exponentiation of an arbitrary element in Fqn with O(n2 loglogn) operations in Fq . We also prove that using a polynomial basis exponentiation in F2n can be done with the same number of operations in F2 for all n. The pre...

Let C be a smooth absolutely irreducible curve of genus g ≥ 1 defined over Fq, the finite field of q elements. Let #C(Fqn) be the number of Fqn -rational points on C. Under a certain multiplicative independence condition on the roots of the zetafunction of C, we derive an asymptotic formula for the number of n = 1, . . . , N such that (#C(Fqn) − q − 1)/2gq belongs to a given interval I ⊆ [−1, 1...