Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients. Asymptotic expression with explicit error bound is derived, which implies that such polynomials of degree 2n always exist provided that the number of prescribed leading coefficients is slightly less than n/4. Exact expressions are also obtained for fields with two or three elements and with up to two prescribed leading coefficients. (C) 2022 Elsevier Inc. All rights reserved.