Some systems of diagonal equations over finite fields
type: Item
instance of: scholarly article
title: Some systems of diagonal equations over finite fields
zbMATH Open document ID: 0910.11015
author: Jacques Wolfmann
published in: Finite Fields and their Applications
publication date: 1999-04-08T00:00:00Z
review text: Let \(A=(a_{i,j})\) be an \((r\times s)\)-matrix of full rank \(r\) with entries in the finite field \(\text{GF}(q)\). For a divisor \(m\) of \(k\) put \(d:=(q^k-1)/(q^m-1)\). The author proves that the number \(N\) of solutions over \(\text{GF}(q^k)\) of the system of equations \[ a_{i,1}x_1^d+\dots +a_{i,s}x_s^d=0 \] is equal to \(Q(d),\) where \(Q(z)\) is the weight polynomial of the orthogonal complement of the linear code generated by \(A\) over \(\text{GF}(q^m)\). This is used to compute \(N\) for some special choices of the matrix \(A\), and the parameters \(s\) and \(r\).
Mathematics Subject Classification ID: 11D79
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 1206731
zbMATH Keywords: systems of homogeneous diagonal equations over finite fields
zbMATH Keywords: congruences in many variables
reviewed by: Hubert Kiechle
MaRDI profile type: Publication
full work available at URL: ffta.1997.0197
OpenAlex ID: W2014894837
cites work: Q3216750
cites work: Q4146667
cites work: The number of solutions of certain diagonal equations over finite fields
cites work: Q4314385
DOI: 10.1006/FFTA.1997.0197
description: scientific article; zbMATH DE number 1206731