The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace
We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of...
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Zusammenfassung: | We present the formula for the number of monic irreducible polynomials of
degree $n$ over the finite field $\mathbb F_q$ where the coefficients of
$x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between
rational points of algebraic curves over finite fields and the number of
elements $a\in\mathbb F_{q^n}$ for which Trace$(a)=0$ and Trace$(a^{-1})=0$.
Besides, we apply the formula to give an upper bound on the number of distinct
constructions of a family of sequences with good family complexity and
cross-correlation measure. |
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DOI: | 10.48550/arxiv.2005.09402 |