Existence of SOLS with holes of type \(2^ n u^ 1\). (English) Zbl 0856.05018
A pair of mutually orthogonal Latin squares (MOLS), where the one square is the transpose of the other, with \(n\) missing sub-MOLS (called holes) of order 2 and one of order \(u\) which are disjoint and spanning, is denoted by \(\text{HSOLS} (2^n u^1)\). D. R. Stinson and L. Zhu [to appear] have shown that an \(\text{HSOLS} (2^n 3^1)\) exists if and only if \(n \geq 4\), except possibly for \(n = 13\) or 15. In this paper, the authors construct an \(\text{HSOLS} (2^n 3^1)\) for \(n = 13\) and 15, and show for \(u = 4\), 5, and 9 that an \(\text{HSOLS} (2^n u^1)\) exists if and only if \(n \geq 1 + u\). They also show for \(u \geq 1\) that an \(\text{HSOLS} (2^n u^1)\) exists if \(n \geq 4 \lceil u/3 \rceil + 10\).
Reviewer: C.J.Salwach (Easton)
MSC:
| 05B15 | Orthogonal arrays, Latin squares, Room squares |
References:
| [1] | Assaf, A.; Mendelsohn, E.; Stinson, D. R., On resolvable coverings of pairs by triples, Utilitas Math., 32, 67-74 (1987) · Zbl 0649.05024 |
| [2] | Bennett, F.; Wu, L.; Zhu, L., Conjugate orthogonal Latin squares with equal-sized holes, Ann. Discrete Math., 34, 65-80 (1987) · Zbl 0628.05011 |
| [3] | Bennett, F.; Wu, L.; Zhu, L., On the existance of COLS with equal-sized holes, Ars. Combin., 26B, 5-36 (1988) · Zbl 0676.05021 |
| [4] | Beth, Th.; Jungnickel, D.; Lenz, H., Design Theory (1985), Bibliographisches Institut: Bibliographisches Institut Zurich · Zbl 0569.05002 |
| [5] | Brayton, R. K.; Coppersmith, D.; Hoffman, A. J., Self-orthogonal Latin squares, (Teore Combinatorie, Proc. Rome Conf. (1976)), 509-517 · Zbl 0363.05018 |
| [7] | Colbourn, C. J.; Hoffman, D. G.; Lindner, C. C., Intersections of \(S(2, 4, v)\) designs, Ars. Combin, 33, 97-111 (1992) · Zbl 0767.05022 |
| [8] | Colbourn, C. J.; Royle, G. F., Supports of \((v, 4, 2)\) designs, Le Matematiche, XLV, 39-60 (1990) · Zbl 0734.05019 |
| [9] | Dinitz, J. H.; Stinson, D. R., MOLS with holes, Discrete Math., 44, 145-154 (1983) · Zbl 0507.05011 |
| [10] | Horton, J. D., Sub-latin squares and incomplete orthogonal arrays, J. Combin. Theory, 16, 23-33 (1974) · Zbl 0297.05015 |
| [11] | Jungnickel, D., Design theory: an update, Ars. Combin, 28, 129-199 (1989) · Zbl 0703.05005 |
| [12] | Lamken, E. R., The existence of three orthogonal partitioned incomplete Latin squares of type \(t^n\), Discrete Math., 89, 231-251 (1991) · Zbl 0726.05024 |
| [13] | Lindner, C. C.; Mullin, R. C.; Stinson, D. R., On the spectrum of resolvable orthogonal arrays invariant under the Klein group \(K_4\), Aequationes Math., 26, 176-183 (1983) · Zbl 0562.05017 |
| [14] | Linder, C. C.; Rodger, C. A., Decomposition into cycles II: cycle systems, (Contemporary Design Theory (1992), Wiley: Wiley New York), 325-369 · Zbl 0774.05078 |
| [15] | Linder, C. C.; Rodger, C. A., 2-perfect \(m\)-cycle systems, Discrete Math., 104, 83-90 (1992) · Zbl 0766.05045 |
| [16] | Mullin, R. C.; Stinson, D. R., Holey SOLSSOMs, Utilitas Math., 25, 159-169 (1984) · Zbl 0553.05021 |
| [17] | Mullin, R. C.; Zhu, L., The spectrum of HSOLSSOM \((h^n)\) where \(h\) is odd, Utilitas Math., 27, 157-168 (1985) · Zbl 0597.05014 |
| [18] | Stinson, D. R.; Zhu, L., On sets of three MOLS with holes, Discrete Math., 54, 321-328 (1985) · Zbl 0572.05017 |
| [19] | Stinson, D. R.; Zhu, L., On the existence of MOLS with equal-sized holes, Aequationes Math., 33, 96-105 (1987) · Zbl 0629.05017 |
| [20] | Stinson, D. R.; Zhu, L., On the existence of three MOLS with equal-sized holes, Australasian J. Combinatorics, 4, 33-47 (1991) · Zbl 0760.05013 |
| [21] | Stinson, D. R.; Zhu, L., On the existence of certain SOLS with holes, JCMCC, 15, 33-45 (1995) · Zbl 0807.05013 |
| [22] | Zhu, L., Existence of holey SOLSSOMs of type \(2^n\), (Congr. Numer., 45 (1984)), 295-304 · Zbl 0589.05030 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
