Certain strongly regular Cayley graphs on F22(2s+1)F22(2s+1) from cyclotomy

We give a construction of negative Latin square type strongly regular Cayley graphs on F22(2s+1)F22(2s+1) with parameters (22(2s+1),2(22s−1)(22s+1+1)/3,4(22s−1)2/9−2,2(22s+1+1)(22s−1)/9)(22(2s+1),2(22s−1)(22s+1+1)/3,4(22s−1)2/9−2,2(22s+1+1)(22s−1)/9) for every s⩾1s⩾1 based on choosing suitable cyclotomic classes on F22(2s+1)F22(2s+1) and the computation of certain Gauss sums.