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Frames with Block Size Four

Dedicated to the memory of Haim Hanani

Published online by Cambridge University Press:  20 November 2018

Rolf S. Rees  and
Douglas R. Stinson
Rolf S. Rees
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NF, Canada A1C 5S7
Douglas R. Stinson
Affiliation:
Department of Computer Science and Engineering, University of Nebraska, Lincoln, NE U.S.A. 68588-0115
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Abstract

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We investigate the spectrum for frames with block size four, and discuss several 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 and only if u ≥ 5, h ≡ 0 mod 3 and h(u — 1) ≡ 0 mod 4, except possibly where

  1. (i) h = 9 and u ∈ ﹛13,17,29,33,93,113,133,153,173,193﹜;

  2. (ii) h ≡ 0 mod 12 and u ∈ ﹛8,12﹜,

  3. h = 36 and u ∈ ﹛7,18,23,28,33,38,43,48﹜,

  4. h = 24 or 120 and u ∈ ﹛7﹜,

  5. h = 72 and u ∈ 2Z+ U ﹛n : n ≡ 3 mod4 and n ≤527﹜ U ﹛563﹜; or

  6. (iii) h ≡ 6mod l2 and u ∈ (﹛17,29,33,563﹜ U ﹛n : n ≡ 3 or 11 mod 12 and n ≤ 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.

Keywords

05B
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 5 , 01 October 1992 , pp. 1030 - 1049
Copyright
Copyright © Canadian Mathematical Society 1992

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