Constant and Nearly Constant Block-Sum Partially Balanced Incomplete Block Designs and Magic Rectangles

We illustrate that the magic rectangles and magic rectangle sets are essentially constant block-sum partially balanced incomplete block designs in disguise. Further, nearly constant block-sum partially balanced incomplete block-sum designs are defined and obtained for the special class when r=1, λ _1 = 1 and λ _2 = 0. It is shown that these are the best designs minimizing the corrected sum of squares due to block-sums. A short catalog of these designs is provided. Thus, these constant or nearly constant block-sum partially balanced incomplete block designs can be constructed for any number of blocks and any block sizes except 2.