arXiv subjects
Publications and source records attributed to Bidushi Sharma.
In this article, we establish a bound on $\theta_q$ that guarantees the existence of a pair of consecutive non-primitive elements in $\mathbb{F}_q$, with the exceptions $q=4$ and $q=8$. We first derive a sufficient condition for the existence of such a pair using character sums and then obtain the stated bound by considering several cases according to the least prime divisor of $q-1$.
In this article, we study polynomials over $\mathbb{F}_{q^2}$ that do not permute $\mathbb{F}_{q^2}$. More precisely, we characterize polynomials of the forms $x^q + b x^2 + c x + d$ and $x^{q+1} + b x^q + c x + d$ over $\mathbb{F}_{q^2}$ according to whether they are permutation or non-permutation polynomials. To this end, we determine the exact number of zeros of these polynomials using existing results on certain special Weil sums.
In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over $\mathbb{F}_{q^{2}}$. The paper focuses on the conditions required for a certain class of degree 2 and degree 3 SCR polynomials to have no roots in $μ_{q+1}$ (the set of $(q+1)-\emph{th}$ roots of unity), which helps in the determination of polynomials that permute $\mathbb{F}_{q^{2}}$. In the due course we also look upon some higher degree SCR polynomials which can be reduced down to a degree 2 SCR polynomial over both odd and even ordered fields. We further look upon the SCR polynomials of type $ax^{q+1}+bx^{q}+bx+a^{q}$ taking both the cases under consideration viz. $a\in \mathbb{F}_{q}$ and $a\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ both.