Some \((17q, 17, 2)\) and \((25q, 25, 3)\) BIBD constructions (Q1299907)

This paper is concerned with the construction of regular balanced incomplete block designs (BIBD) of the series \((17q, 17, 2)\) and \((25q, 25, 3)\) with some constraints placed on \(q\). The following are the two main results of this paper. Result I. Let \(q\equiv 17\pmod{32}\) be a prime power such that 2 is not a 4th power in \(\text{GF}(q)\). Then there exists a regular \((17q, 17, 2)\)-BIBD. Result II. If \(q\equiv 25\pmod{48}\) is a prime power such that \(\sqrt 3\) and \(\sqrt 3+3\) are non-squares in \(\text{GF}(q)\), there exists a regular \((25q, 25, 3)\)-BIBD. Using Result I, the author claims to have found a cyclic \((17q, 17, 2)\)-BIBD for each prime \(q= 241,401,433,\dots, 4337,4561\), and Result II leads to a regular \((25q, 25, 3)\)-BIBD for each \(q= 25,409,457,\dots, 9769,9817\).