The spectrum \(Q(k, \lambda)\) of coset difference arrays with \(k = 2\lambda+ 1\)

The spectrum \(Q(k,\lambda)\) of coset difference arrays has played an important role in Lu’s work on asymptotic existence of resolvable balanced incomplete block designs. In this article, we use Weil’s theorem on character sums to show that if \(k = 2\lambda + 1\), then for any prime power \(q \equiv 1+2k \pmod{4k}\), \(q \in Q(k,\lambda)\) whenever \(g > D(k) = (\frac{B+\sqrt{B^2+4C}}{2})^2\), where \(B = (k-2)k(2k-1)(2k)^{k-1} – (2k)^{k} + 1\) and \(C = \frac{(k-2)(k-1)}{2}(2k)^{k-1}\). In particular, we determine the spectrum \(Q(3,1)\). In addition, the degenerate case when \(k = \lambda + 1\) is also discussed.