Frames with Block Size Four

Rees, Rolf S. ; Stinson, Douglas R.

Canadian journal of mathematics, Tome 44 (1992) no. 5, pp. 1030-1049

Résumé

We investigate the spectrum for frames with block size four, and discussseveral applications to the construction of other combinatorial designs.Our main result is that a frame of type hu, having blocks of size four, exists if andonly if u ≥ 5, h ≡ 0 mod 3 and h(u — 1) ≡ 0 mod 4, except possibly where (i) h = 9 and u ∈ {13,17,29,33,93,113,133,153,173,193}; (ii) h ≡ 0 mod 12 and u ∈{8,12}, h = 36 and u ∈ {7,18,23,28,33,38,43,48}, h = 24 or 120 and u ∈{7}, h = 72 and u ∈2Z+ U {n : n ≡ 3 mod4 and n ≤527} U {563}; or (iii) h ≡ 6mod l2 and u ∈({17,29,33,563} U {n : n ≡ 3 or 11 mod 12 andn ≤ 527} U {n : n ≡ 7 mod 12 and n ≤ 259}), h = 18. Additionally, we give a new recursive construction for resolvable group-divisible designs from frames: if there is a resolvable k-GDD of type gu, a k-frame of type {mg)v where u ≥ m + 1, and a resolvable TD(k, mv) then there is a resolvable k-GDD of type (mg)uv. We use this to construct some new resolvable GDDs with group size three and block size four.

Détail

DOI : 10.4153/CJM-1992-063-x

Mots-clés : 05B

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