scholarly journals Class-Uniformly Resolvable Group Divisible Structures II: Frames.

10.37236/1777 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Peter Danziger ◽  
Brett Stevens
Keyword(s):  
Blow Up ◽  
Necessary Conditions ◽  
Group Divisible Designs ◽  
Partial Resolution ◽  
Group Divisible ◽  
Uniform Condition ◽  
Resolvable Group ◽  
Closure Result ◽  
General Curf

We consider Class-Uniformly Resolvable frames (CURFs), which are group divisible designs with partial resolution classes subject to the class-uniform condition. We derive the necessary conditions, including extremal bounds, build the foundation for general CURF constructions, including a frame variant of the $\lambda$ blow-up construction from part I. We also establish a PBD-closure result. For CURFs with blocks of size two and three we determine the existence of CURFs of type $g^u$, completely for $g=3$, with a small list of exceptions for $g=6$, asymptotically for $g=4,5$ and give some other infinite families.

scholarly journals Class-Uniformly Resolvable Group Divisible Structures I: Resolvable Group Divisible Designs

10.37236/1776 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Peter Danziger ◽  
Brett Stevens
Keyword(s):  
Necessary Conditions ◽  
Group Divisible Designs ◽  
Group Divisible ◽  
Master Design ◽  
Block Sizes ◽  
Resolvable Group

We consider Class-Uniformly Resolvable Group Divisible Designs (CURGDD), which are resolvable group divisible designs in which each of the resolution classes has the same number of blocks of each size. We derive the fully general necessary conditions including a number of extremal bounds. We present some general constructions including a novel construction for shrinking the index of a master design. We construct a number of infinite families, primarily with block sizes 2 and $k$, including some extremal cases.

scholarly journals LDPC Codes Based on α – Resolvable BIB and Group Divisible Designs

2021 ◽  
Author(s):  
Shyam Saurabh
Keyword(s):  
Block Design ◽  
Ldpc Codes ◽  
Ldpc Code ◽  
Finite Geometries ◽  
Group Divisible Designs ◽  
Tanner Graph ◽  
Group Divisible ◽  
Check Matrix ◽  
Balanced Incomplete Block ◽  
Resolvable Group

<p>Structured LDPC codes have been constructed using balanced incomplete block (BIB) designs, resolvable BIB designs, mutually orthogonal Latin rectangles, partial geometries, group divisible designs, resolvable group divisible designs and finite geometries. Here we have constructed LDPC codes from <i>α </i>–<b> </b>resolvable BIB and Group divisible designs. The sub–matrices of incidence matrix of such block design are used as a parity – check matrix of the code which satisfy row – column constraint. Here the girth of the proposed code is at least six and the corresponding LDPC code (or Tanner graph) is free of 4– cycles. </p>

Frames with Block Size Four

1992 ◽  
Vol 44 (5) ◽  
pp. 1030-1049 ◽  
Author(s):  
Rolf S. Rees ◽  
Douglas R. Stinson
Keyword(s):  
Group Size ◽  
Block Size ◽  
List Type ◽  
Combinatorial Designs ◽  
Recursive Construction ◽  
Type Number ◽  
Group Divisible Designs ◽  
Group Divisible ◽  
Resolvable Group

AbstractWe 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(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 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.

scholarly journals Resolvable group divisible designs with block size 3

1989 ◽  
Vol 77 (1-3) ◽  
pp. 5-20 ◽  
Author(s):  
Ahmed M. Assaf ◽  
Alan Hartman
Keyword(s):  
Block Size ◽  
Group Divisible Designs ◽  
Group Divisible ◽  
Resolvable Group
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