Nested balanced incomplete block designs.

Abstract If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v,b 1 ,r,k 1 ) are each partitioned into sub-blocks of size k 2 , and the b 2 =b 1 k 1 /k 2 sub-blocks themselves constitute a BIBD with parameters (v,b 2 ,r,k 2 ) , then the system of blocks, sub-blocks and treatments is, by definition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k 1 =2k 2 =4 . Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, they are almost unknown in the combinatorial literature, except in the literature of tournaments, and detailed combinatorial studies of them have been lacking. The present paper therefore reviews and extends mathematical knowledge of NBIBDs. Isomorphism and automorphisms are defined for NBIBDs, and methods of construction are outlined. Some special types of NBIBD are defined and illustrated. A first-ever detailed table of NBIBDs with v⩽16 , r⩽30 is provided; this table contains many newly discovered NBIBDs.

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