Abstract
An (m, n; u, v; c)-system is a collection of components,m of valencyu − 1 andn of valencyv − 1, whose difference sets form a perfect system with thresholdc. A necessary condition for the existence of an (m, n; u, v; c)-system foru = 3 or 4 is thatm ≥ 2c − 1; and there are (2c − 1,n; 3, 6;c)-systems for all sufficiently largec at least whenn = 1 or 2. It is shown here that if there is a (2c − 1,n; u, 6;c)-system thenn = 0 whenu = 4 andn ≤ 2c − 1 whenu = 3. Moreover, if there is a (2c − 1,n; 3, 6;c)-system with a certain splitting property thenn ≤ c − 1, this last result being of possible interest in connection with the multiplication theorem for perfect systems.
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Rogers, D.G. Irregular, extremal perfect systems of difference sets II. Graphs and Combinatorics 4, 285–295 (1988). https://doi.org/10.1007/BF01864167
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DOI: https://doi.org/10.1007/BF01864167