Self-reciprocal irreducible polynomials over finite fields
From the introduction: Let \(\mathbb F_q\) denote the finite field containing \(q\) elements, where \(q=p^e\) is a prime power. In the following polynomial will be understood to mean a monic polynomial over \(\mathbb F_q\). For \(f(x)\) a polynomial of degree \(n\) over \(\mathbb F_q\), the reciprocal of \(f(x)\) is the polynomial \(f^*(x)\) of degree \(n\) over \(\mathbb F_q\) given by \(f^*(x)=x^n f (1/x)\), and a polynomial \(f(x)\) is called self-reciprocal if \(f(x)= f^*(x)\). We note that if \(f(x)\) is irreducible (primitive) over \(\mathbb F_q\), then so is the reciprocal polynomial \(f^*(x)\). Numerous results have been obtained concerning self-reciprocal irreducible polynomials over finite fields; see, for example, Section 2.7 of Jungnickel's monograph [Finite fields: structure and arithmetics. Mannheim: B. I.-Wissenschaftsverlag (1993; Zbl 0779.11058)] and \textit{H. Meyn} [Appl. Algebra Eng. Commun. Comput. 1, No. 1, 43--53 (1990; Zbl 0724.11062)]. Self-reciprocal polynomials over finite fields are useful in several applications including reversible codes with read-backward properties and primitive self-complementary necklaces which consist of beads with \(q\) colors. An irreducible polynomial of degree at least two is self-reciprocal if and only if its set of roots is closed under inversion, so its degree must be even. As indicated in Theorem 2.7.4 of Jungnickel's monograph, the number of self-reciprocal irreducible polynomials of degree \(n=2m\) over \(\mathbb F_q\) is given by \(1=2mq^{m-1}\) if \(q\) is odd and \(m=2^a\), and \(\frac {1}{2m}\sum_{d\mid m, d \text{ odd}}\) otherwise. The order of a polynomial \(f\) over \(\mathbb F_q\) is the smallest positive integer \(s\) so that \(f(x)\) divides \(x^s-1\). The order of a polynomial \(f\) will be denoted by \(\text{ord}(f)\). If \(f\) is irreducible of degree \(n\), then \(\text{ord}(f)\) is the order of any root of \(f\) in the multiplicative group \(\mathbb F^*_{q^n}\). In Section 2 of this paper, the authors determine the orders of all self-reciprocal irreducible polynomials over \(\mathbb F_q\). This leads to the main result, a classification of the self-reciprocal irreducible polynomials (Theorem 8). They also present a different count of the number of self-reciprocal irreducible polynomials and a factorization of certain cyclotomic polynomials over \(\mathbb F_q\). In Section 3, they study the weights of self-reciprocal irreducible polynomials.
scientific article; zbMATH DE number 2128563