Hadamard tournaments with transitive automorphism groups (Q793038)

A THT is a Hadamard \(2-(4d+3,2d+1,d)\) design \(D=(P,B)\) with a transitive automorphism group G. Classic examples of THT are quadratic residue (or non residue) designs D(q), where q is a prime power such that \(q\equiv 3(mod 4)\). \textit{J. B. Kelly} [Proc. Am. Math. Soc. 5, 38-46 (1954; Zbl 0055.037)] and \textit{E. C. Johnsen} [J. Algebra 4, 388-402 (1966; Zbl 0178.335)], independently, characterized D(q) with q a prime as a THT such that G contains a cyclic transitive subgroup. \textit{C. W. Lam} [Discrete Math. 29, 265-274 (1980; Zbl 0442.05021)] characterized D(q) as a THT such that G is of rank 3. In this paper the author observes that the previous assumptions imply \(4d+3\) is a prime power and he proves that this fact is true for large classes of THT.