On the existence of (400, 57, 8) non-abelian difference sets

Difference sets with parameters (frac{q^{d + 1} - 1}{q - 1}, frac{q^d - 1}{q - 1}, frac{q^{d - 1} - 1}{q - 1}), where q is a prime power and d geq 1, are known to exist in cyclic groups and are called classical Singer difference sets. We study a special case of this family with q = 7 and d = 3 in search of more difference sets. According to GAP, there are 220 groups of order 400 out of which 10 are abelian. E. Kopilovich and other authors showed that the remaining nine abelian groups of order 400 do not admit (400, 57, 8) difference sets. Also, Gao and Wei used the (400, 57, 8) Singer difference set to construct four inequivalent difference sets in a non-abelian group. In this paper, we demonstrate using group representation and factorization in cyclotomic rings that, out of the remaining 209 non-abelian groups of order 400, only 15 could possibly admit (400, 57, 8) difference sets.