The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace
This work addresses a specific counting problem in finite field theory, with incremental contributions to sequence design in coding or cryptography.
The authors derived a formula for counting monic irreducible polynomials over finite fields with vanishing trace and reciprocal trace coefficients, relating this to rational points on algebraic curves, and applied it to bound constructions of sequences with good complexity and correlation measures.
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.