The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

TL;DR

This paper derives a formula for counting certain irreducible polynomials over finite fields with specific coefficient conditions, linking algebraic curves and sequence construction metrics.

Contribution

It introduces a new formula connecting polynomial enumeration with algebraic curve points and sequence complexity measures.

Findings

Derived a formula for counting irreducible polynomials with vanishing trace and reciprocal trace.

Established a relation between algebraic curve points and elements with specific trace properties.

Provided an upper bound on sequence construction complexity and cross-correlation.

Abstract

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.