×

zbMATH — the first resource for mathematics

Designs with mutually orthogonal resolutions. (English) Zbl 0618.05009
A combinatorial design D with replication number r is said to be resolvable if the blocks of D can be partitioned into classes \(R_ 1,R_ 2,...,R_ r\) such that each element of D is contained in precisely one block of each class. Two resolutions R and R’ of D are called orthogonal if \(| R_ i\cap R_ j'| \leq 1\) for all \(R_ i\in R\), \(R_ j'\in R'\). A set \(Q=\{R^ 1,R^ 2,...,R^ t\}\) of t resolutions of D is called a set of mutually orthogonal resolutions (MORs) if the resolutions of Q are pairwise orthogonal. The authors construct designs with sets of t MORs for several sequences of t. Furthermore, for given designs they determine upper bounds for t.
Reviewer: H.Groh

MSC:
05B05 Combinatorial aspects of block designs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, B.A., Hyperovals and howell designs, Ars combin., 9, 29-38, (1980) · Zbl 0456.05021
[2] Bondy, J.A.; Murty, U.S. R., Graph theory with application, (1976), Basingstore MacMillan London · Zbl 1226.05083
[3] Deza, M.; Mullin, R.C.; Vanstone, S.A., Orthogonal systems, Aequationes math., 17, 322-330, (1978) · Zbl 0377.05012
[4] Dinitz, J., New lower bounds for the number of pairwise orthogonal symmetric Latin squares, Congressus numerantium (1), 22, 393-398, (1979)
[5] Dinitz, J.; Stinson, D.R., The spectrum of room cubes, Europ. J. combinatorias (3), 2, 221-230, (1981) · Zbl 0531.05014
[6] Fuji-Hara, R.; Vanstone, S.A., On the spectrum of doubly resolvable designs, Congressus numerantium, 28, 399-407, (1980) · Zbl 0456.05020
[7] Hughes, D.R.; Piper, F.C., Projective planes, Springer-verlag, (1973) · Zbl 0267.50018
[8] Lamken, E.R.; Vanstone, S.A., Elliptic semiplanes and group divisible designs with orthogonal resolutions, Aequationes math.,, 30, 80-92, (1986) · Zbl 0586.05006
[9] R. Mathon and P. Gibbons, Construction methods for Bhaskar Rao and related Designs, J. of Australia Math. Soc. (to appear). · Zbl 0638.05006
[10] Mullin, R.C.; Wallis, W.D., The existence of room squares, Aequationes math., 13, 1-7, (1975) · Zbl 0315.05014
[11] Rosa, A.; Vanstone, S.A., Starter-adder techniques for kirkman squares and kirkman cubes of small sides, Ars combin., 14, 199-212, (1982) · Zbl 0506.05013
[12] Ryser, H.J., Combinatorial mathematics, Carus math. monograph, no. 14, (1963), MAA, Wiley New York · Zbl 0112.24806
[13] Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. amer. math. soc., 43, 377-385, (1938) · JFM 64.0972.04
[14] Vanstone, S.A., Doubly resolvable designs, Discrete math., 29, 77-86, (1980) · Zbl 0447.05010
[15] Vanstone, S.A., On mutually orthogonal resolutions and near-resolutions, Annals of discrete math, 15, 357-369, (1982) · Zbl 0512.05007
[16] S. A. Vanstone, unpublished.
[17] Vanstone, S.A.; Schellenberg, P.J., A construction for equidistant permutation arrays of index one, J. combin. theory ser. A, 23, 180-186, (1977) · Zbl 0361.05026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.