Existence of \((q,6,1)\) difference families with \(q\) a prime power (Q1273541)

A family \(\mathcal F\) of \(k\)-subsets of an additive group \(G\) of order \(v\) is called a \((v,k,1)\)-DF if any nonzero element of \(G\) can be represented in a unique way as difference of two elements of some member of \(\mathcal F\). If \(G\) is the additive group of \(\text{GF}(q)\) the necessary condition \(q\equiv 1\pmod {30}\) is also sufficient with the exception \(q=61\) only. The result is obtained using Wilson's ideas, character sums and computer search for \(q\in[10000,360191]\).