论文信息 - On the existence of frames

On the existence of frames

This paper deals with two topics, namely, frames and pairwise balanced designs (PBD's). Frames, which were introduced by W.D. Wallis for the construction of (skew) Room squares, are shown to exist for most orders congruent to 1 (mod 4). This result relies heavily on the existence of PBD's since the set F = {v | there is a frame of order v] is shown to be PBD-closed. By employing a generalization of the usual recursive construction for PBD's, it is shown that B{5, 9, 13, 17}@?B{5, 9, 13}@?{69, 77, 97, 137, 237, 277, 317, 377, 569}@?{n | n @? 1 (mod 4), n>0}@?{29, 33, 49, 57, 93, 129, 133}, where B(K) denotes the set of orders of PBD's of index one having block-sizes from the set K. Frames of orders 5, 9, 13 and 17 are exhibited which immediately implies that F@?B{5, 9, 13, 17}. D.R. Stinson and W.D. Wallis have shown that {29, 49}@?F. Thus there is a frame of order @u for every positive integer @u congruent to 1 (mod 4) with the possible exceptions of @u @e {33, 57, 93, 133}.

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