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Nearly Kirkman Triple Matching

A Nearly Kirkman Triple System is a \(\left(3, 1\right)\)-\(\mathrm{RGDD}\) of type \(2^{n}\). Baker and Wilson (1977)1 showed that such systems exists where the total number of points is greater than or equal to 18 and is divisible by 6. This means that \(l_{\max}(\alpha, 3)\) is equal to the trivial upper bound when \(\alpha \geq 6\) and is even. The non existence of systems for fewer points also implies that \(l_{\max}(4, 3)\) is less than the trivial upper bound.

Constructions

The perfect-strangers package currently implements the nearly Kirkman triple systems given in Table I of Abel et al. (2013)2.

  1. Baker, R.D. and Wilson, R.M., 1977. Nearly Kirkman triple systems. Utilitas Math, 11, pp.289-296. ↩

  2. Abel, R.J.R., Chan, N., Colbourn, C.J., Lamken, E.R., Wang, C. and Wang, J., 2013. Doubly resolvable nearly Kirkman triple systems. Journal of Combinatorial Designs, 21(8), pp.342-358. DOI: 10.1002/jcd.21342 ↩