Self-reciprocal irreducible polynomials over finite fields
type: Item
instance of: scholarly article
title: Self-reciprocal irreducible polynomials over finite fields
zbMATH Open document ID: 1146.11336
author: Joseph L. Yucas
author: Gary L. Mullen
published in: Designs, Codes and Cryptography
publication date: 2005-01-17T00:00:00Z
review text: 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,…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 2128563
zbMATH Keywords: Finite field
zbMATH Keywords: self-reciprocal
zbMATH Keywords: polynomial
MaRDI profile type: Publication
full work available at URL: b:desi.0000036251.41345.1f
OpenAlex ID: W1969088895
DOI: 10.1023/B:DESI.0000036251.41345.1F
description: scientific article; zbMATH DE number 2128563