A correction on the determination of the weight enumerator polynomial of some irreducible cyclic codes

A classification that shows explicitly all possible weight enumerator polynomials for every irreducible cyclic code of length n over a finite field Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document}, in the particular case where each prime divisor of n is also a divisor of q-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q-1$$\end{document}, was recently given in Brochero Martínez and Giraldo Vergara (Des Codes Cryptogr 78:703–712, 2016). However, as we will see next, such classification is incomplete. Thus, the purpose of this work is to use an already known identity among the weight enumerator polynomials, in order to complete such classification. As we will see later, by means of this identity, we not only complete, in an easier way, this classification, but we also find out the nature of the weight distributions of the class of irreducible cyclic codes studied in Brochero Martínez and Giraldo Vergara (2016).