Resolvable balanced incomplete block designs with subdesigns of block size 4 Original Research Article

In this paper, we look at resolvable balanced incomplete block designs on image points having blocks of size 4, briefly image RBIBDs. The problem we investigate is the existence of image RBIBDs containing a image RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an image. The necessary conditions for the existence of an image are that image and image. We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever image and whenever image. We also give some results on pairwise balanced designs on image points containing (at least one) block of size image, i.e., a image-PBD. If the list of permitted block sizes, image, contains all integers of size 5 or more, and image, then a necessary condition on this PBD is image. We show this condition is not sufficient for any image and give the complete spectrum (in image) for image, as well as showing the condition image is sufficient with some definite exceptions for image and image, and some possible exceptions when image, namely image. The existence of this PBD implies the existence of an image. If the list of permitted block sizes, image, contains all integers image, and image, then a necessary condition on this PBD is image. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when image. The existence of this PBD implies the existence of an IRBimage.