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.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

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

VL - 26

SP - 169

EP - 196

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 1-3

ER -