Authors: Tomas Nilson Pia Heidtmann
Publish Date: 2012/07/22
Volume: 71, Issue: 2, Pages: 247-260
Abstract
A triple array is a rowcolumn design which carries two balanced incomplete block designs BIBDs as substructures McSorley et al Des Codes Cryptogr 35 21–45 2005 Section 8 gave one example of a triple array that also carries a third BIBD formed by its rowcolumn intersections This triple array was said to be balanced for intersection and they made a search for more such triple arrays among all potential parameter sets up to some limit No more examples were found but some candidates with suitable parameters were suggested We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced As triple arrays in the canonical case correspond to SBIBDs this in turn yields new existence criteria for triple arrays balanced for intersection In particular we prove that the residual design of the related SBIBD with respect to the defining block must be quasisymmetric and give necessary and sufficient conditions on the intersection numbers This together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al Des Codes Cryptogr 35 21–45 2005 and many others in a wide search Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block We show as a key result that such SBIBDs must possess the quasi3 property and we answer the existence question for all known classes of these designs
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