A few more Kirkman squares and doubly near resolvable BIBDs with block size 3 Original Research Article

A Kirkman square with index image, latinicity image, block size k, and image points, image, is a image array (image) defined on a image-set V such that (1) every point of V is contained in precisely image cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a image-BIBD. In a series of papers, Lamken established the existence of the following designs: image with at most six possible exceptions [E.R. Lamken, The existence of doubly resolvable image-BIBDs, J. Combin. Theory Ser. A 72 (1995) 50–76], image with two possible exceptions [E.R. Lamken, The existence of image, Discrete Math. 186 (1998) 195–216], and doubly near resolvable image-BIBDs with at most eight possible exceptions [E.R. Lamken, The existence of doubly near resolvable image-BIBDs, J. Combin. Designs 2 (1994) 427–440]. In this paper, we construct designs for all of the open cases and complete the spectrum for these three types of designs. In addition, Colbourn, Lamken, Ling, and Mills established the spectrum of image in 2002 with 23 possible exceptions. We construct designs for 11 of the 23 open cases.