Some systems of diagonal equations over finite fields (Q1267002)

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 \(r\).