# Recursive Methods for Construction of Balanced N-ary Block Designs

• Volume: 31, Issue: 3, page 189-200
• ISSN: 1310-6600

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## Abstract

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2000 Mathematics Subject Classification: Primary 05B05; secondary 62K10.This paper presents a recursive method for the construction of balanced n-ary block designs. This method is based on the analogy between a balanced incomplete binary block design (B.I .E .B) and the set of all distinct linear sub-varieties of the same dimension extracted from a finite projective geometry. If V1 is the first B.I .E .B resulting from this projective geometry, then by regarding any block of V1 as a projective geometry, we obtain another system of B.I .E .B. Then, by reproducing this operation a finite number of times, we get a family of blocks made up of all obtained B.I .E .B blocks. The family being partially ordered, we can obtain an n-ary design in which the blocks are consisted by the juxtaposition of all binary blocks completely nested. These n-ary designs are balanced and have well defined parameters. Moreover, a particular balanced n-ary class is deduced with an appreciable reduction of the number of blocks.

## How to cite

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Gheribi-Aoulmi, Z., and Bousseboua, M.. "Recursive Methods for Construction of Balanced N-ary Block Designs." Serdica Mathematical Journal 31.3 (2005): 189-200. <http://eudml.org/doc/219563>.

@article{Gheribi2005,
abstract = {2000 Mathematics Subject Classification: Primary 05B05; secondary 62K10.This paper presents a recursive method for the construction of balanced n-ary block designs. This method is based on the analogy between a balanced incomplete binary block design (B.I .E .B) and the set of all distinct linear sub-varieties of the same dimension extracted from a finite projective geometry. If V1 is the first B.I .E .B resulting from this projective geometry, then by regarding any block of V1 as a projective geometry, we obtain another system of B.I .E .B. Then, by reproducing this operation a finite number of times, we get a family of blocks made up of all obtained B.I .E .B blocks. The family being partially ordered, we can obtain an n-ary design in which the blocks are consisted by the juxtaposition of all binary blocks completely nested. These n-ary designs are balanced and have well defined parameters. Moreover, a particular balanced n-ary class is deduced with an appreciable reduction of the number of blocks.},
author = {Gheribi-Aoulmi, Z., Bousseboua, M.},
journal = {Serdica Mathematical Journal},
keywords = {Balanced Incomplete Binary Blocks; N-ary Designs; Finite Projective Geometry; Finite Linear Sub-Variety; finite projective geometries},
language = {eng},
number = {3},
pages = {189-200},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Recursive Methods for Construction of Balanced N-ary Block Designs},
url = {http://eudml.org/doc/219563},
volume = {31},
year = {2005},
}

TY - JOUR
AU - Gheribi-Aoulmi, Z.
AU - Bousseboua, M.
TI - Recursive Methods for Construction of Balanced N-ary Block Designs
JO - Serdica Mathematical Journal
PY - 2005
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 31
IS - 3
SP - 189
EP - 200
AB - 2000 Mathematics Subject Classification: Primary 05B05; secondary 62K10.This paper presents a recursive method for the construction of balanced n-ary block designs. This method is based on the analogy between a balanced incomplete binary block design (B.I .E .B) and the set of all distinct linear sub-varieties of the same dimension extracted from a finite projective geometry. If V1 is the first B.I .E .B resulting from this projective geometry, then by regarding any block of V1 as a projective geometry, we obtain another system of B.I .E .B. Then, by reproducing this operation a finite number of times, we get a family of blocks made up of all obtained B.I .E .B blocks. The family being partially ordered, we can obtain an n-ary design in which the blocks are consisted by the juxtaposition of all binary blocks completely nested. These n-ary designs are balanced and have well defined parameters. Moreover, a particular balanced n-ary class is deduced with an appreciable reduction of the number of blocks.
LA - eng
KW - Balanced Incomplete Binary Blocks; N-ary Designs; Finite Projective Geometry; Finite Linear Sub-Variety; finite projective geometries
UR - http://eudml.org/doc/219563
ER -

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