On (\(g\), 4; 1)-difference matrices (Q2581625)

Let \(G\) be an abelian group of order \(g\) and \(A=(a_{ij})\) be a \(k\times g\) matrix with enteries in \(G\). If the differences \(a_{rj}-a_{sj}\), \(1\leq j\leq g\), when \(r\) and \(s\) range according to \(1\leq r<s\leq k\), comprise all the elements of \(G\), then \(A\) is called a \((g,k;1)\)-difference matrix based on \(G\). In the present paper the author proves that a \((g,4;1)\)-difference matrix exists if and only if \(g\geq 4\) and \(g\not\equiv 2\pmod 4\).