Abstract
It has been conjectured by Dinesh Sarvate and Jennifer Seberry, that, when p is an odd prime or prime power congruent to 1 mod 4, a certain collection of p sets of p elements can be used to define uniquely an SBIBD(2p+ 1, p, 1/2(p-1)), and that, when p is a prime power congruent to 3 mod 4, then a certain collection of 1/2(p-1) sets can be used to define uniquely an SBIBD(p, 4(p - 1), 1/4(p - 3)). This would mean that, in certain cases, 2t - 1 rows are enough to complete uniquely the Hadamard matrix of order 4t. Examples suggest that the defining sets can be used first to find the corresponding residual BIBD, which can then be extended uniquely to the SBIBD. These conjectures have now been verified for p = 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 43, 47, 59 and 67, providing further examples in which a residual design with λ > 2 is completable to an SBIBD, the first such case having been given in Seberry in 1992.
Publication Details
Thomas Kunkle and Jennifer Seberry, A few more small defining sets for SBIBD(4t-1, 2t-1, t-1), Bulletin of the Institute of Combinatorics and its Applications, 12, (1994), 61-64.