An asymptotic result on the existence of generalised Hadamard matrices (Q1317462)

The authors prove the following important asymptotic existence result for generalised Hadamard matrices: Given an ordinary Hadamard matrix of order \(k\), there exists a generalised Hadamard matrix of size \(kq \times kq\) over the elementary abelian group of order \(q\) whenever \(q\) is a sufficiently large odd prime power. The proof uses character sums over finite fields to establish a certain pseudo-randomness property of the quadratic character of \(\text{GF} (q)\) which is exploited in the construction of the desired GH-matrix.