Counting polynomials with distinct zeros in finite fields

Abstract

Let F q be a finite field with q = p e elements, where p is a prime and e ≥

On the enumeration of polynomials with prescribed factorization pattern

“…Corollary 16 (Theorem 3.1 [20]). The number of monic polynomials over F q of the form x m + αx m−1 + g(x) for fixed α ∈ F q , where g ∈ F q [x] has degree at most m − 2, that have r distinct linear factors is given as follows:…”

“…, f w , where w ≥ 1 is arbitrary, due to applications in Reed-Solomon codes (see [20,10]). For the case w = 1, Zhou et al [20] studied the number of degree m ≥ 1 polynomials over F q with r distinct roots of the form x m +αx m−1 +g(x), where α ∈ F q is fixed, and g(x) ∈ F q [x] is a varying polynomial of degree at most m − 2. If p ∤ m, then the number is…”

“…Here we use the notation P , which is equal to 1 if P is true and P = 0 otherwise. The exact and more complicated expressions are obtained in [20] for the number of monic degree m ≥ 2 polynomials with r distinct linear factors over F q of the form x m +g(x) where g(x) ∈ F q [x] is a varying polynomial of degree at most m−3. More recently, in [10], an asymptotic bound on the number of degree m ≥ w polynomials with r distinct linear factors over F q of the form…”

Distance Distribution in Reed-Solomon Codes

“…where v(a) is a small constant depending on a. When m = 2, then deg(f ) = k + 2 and an explicit but complicated counting formula is also given in [16]. Details are omitted.…”

“…It turns out that N (x k+1 + ax k , r) depends on a. It was proved by Zhou, Wang and Wang [16] that when p ∤ k + 1, then for 0 ≤ r ≤ k + 1,…”