The Existence of Kirkman Squares—Doubly Resolvable (v,3,1)-BIBDs

  • Authors:

  • Charles J. Colbourn;E. R. Lamken;Alan C. H. Ling;W. H. Mills

  • Affiliations:

  • Computer Science and Engineering, Arizona State University, Tempe AZ 85287-5406, U.S.A. Charles.Colbourn@asu.edu;Mathematics, 253-37, California Institute of Technology, California, CA 91125, U.S.A. lamken@its.caltech.edu;Computer Science, University of Vermont, Burlington, VT 05405, U.S.A. aling@emba.uvm.edu;CCR-IDA, Thanet Road, Princeton, NJ 08540, U.S.A.

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2002

Quantified Score

Hi-index 0.00

Visualization

Abstract

A Kirkman square with index λ, latinicity μ, block size k, and v points, KSk(v;μ,λ), is a t×t array (t=λ(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,λ)-BIBD. For μ=1, the existence of a KSk(v; μ, λ) is equivalent to the existence of a doubly resolvable (v, k, λ)-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 λ=1. We show that there exist KS3 (v; 1, 1) for v \equiv 3 \pmod 6, v=3 and v≥27 with at most 23 possible exceptions for v.