Uniformly resolvable designs with block sizes 3 and 4

Hengjia Wei; Gennian Ge

doi:10.1016/j.disc.2015.10.042

Uniformly resolvable designs with block sizes 3 and 4

Hengjia Wei, Gennian Ge

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k. Such a class is denoted k-pc and for a given k the number of k-pcs is denoted r_k. Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0(mod12). It has been shown that almost all URDs with permissible r₃ and r₄ exist for v ≡ 0(mod24), v ≡ 0(mod60), v ≡ 36(mod144) or v ≡ 36(mod108). In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v=12, r₃=1 and r₄=3.

Original language	English
Pages (from-to)	1069-1085
Number of pages	17
Journal	Discrete Mathematics
Volume	339
Issue number	3
DOIs	https://doi.org/10.1016/j.disc.2015.10.042
State	Published - 6 Mar 2016
Externally published	Yes

Keywords

Frame
Resolvable group divisible design
Uniformly resolvable design

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/j.disc.2015.10.042

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title = "Uniformly resolvable designs with block sizes 3 and 4",

abstract = "A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k. Such a class is denoted k-pc and for a given k the number of k-pcs is denoted rk. Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0(mod12). It has been shown that almost all URDs with permissible r3 and r4 exist for v ≡ 0(mod24), v ≡ 0(mod60), v ≡ 36(mod144) or v ≡ 36(mod108). In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v=12, r3=1 and r4=3.",

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AU - Wei, Hengjia

AU - Ge, Gennian

PY - 2016/3/6

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N2 - A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k. Such a class is denoted k-pc and for a given k the number of k-pcs is denoted rk. Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0(mod12). It has been shown that almost all URDs with permissible r3 and r4 exist for v ≡ 0(mod24), v ≡ 0(mod60), v ≡ 36(mod144) or v ≡ 36(mod108). In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v=12, r3=1 and r4=3.

AB - A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k. Such a class is denoted k-pc and for a given k the number of k-pcs is denoted rk. Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ≡ 0(mod12). It has been shown that almost all URDs with permissible r3 and r4 exist for v ≡ 0(mod24), v ≡ 0(mod60), v ≡ 36(mod144) or v ≡ 36(mod108). In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v=12, r3=1 and r4=3.

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KW - Resolvable group divisible design

KW - Uniformly resolvable design

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Uniformly resolvable designs with block sizes 3 and 4

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Keywords

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