TY - JOUR
T1 - The existence of Kirkman squares-doubly resolvable (v, 3, 1)-BIBDs
AU - Colbourn, Charles
AU - Lamken, E. R.
AU - Ling, Alan C H
AU - Mills, W. H.
N1 - Publisher Copyright:
© 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
PY - 2002
Y1 - 2002
N2 - A Kirkman square with index γ, latinicity μ, block size k, and v points, KSk(v; μ, γ), is a t × t array (t = γ(v - 1)/μ(k - 1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k, γ)-BIBD. For μ = 1, the existence of a KSk (v; μ, γ) is equivalent to the existence of a doubly resolvable (v, k, γ)-BIBD. The spectrum of KS2(v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with γ = 1. We show that there exist KS3(v; 1, 1) for v ≡ 3 (mod 6), v = 3 and v ≥ 27 with at most 23 possible exceptions for v.
AB - A Kirkman square with index γ, latinicity μ, block size k, and v points, KSk(v; μ, γ), is a t × t array (t = γ(v - 1)/μ(k - 1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k, γ)-BIBD. For μ = 1, the existence of a KSk (v; μ, γ) is equivalent to the existence of a doubly resolvable (v, k, γ)-BIBD. The spectrum of KS2(v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with γ = 1. We show that there exist KS3(v; 1, 1) for v ≡ 3 (mod 6), v = 3 and v ≥ 27 with at most 23 possible exceptions for v.
KW - Doubly resolvable
KW - Kirkman square
KW - Kirkman triple system
KW - Resolvable
KW - Steiner triple system
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U2 - 10.1023/a:1016513527747
DO - 10.1023/a:1016513527747
M3 - Article
AN - SCOPUS:38149037292
SN - 0925-1022
VL - 26
SP - 169
EP - 196
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
IS - 1-3
ER -