Designs 2002 pp 23-45

# Conjugate Orthogonal Diagonal Latin Squares with Missing Subsquares

• Frank E. Bennett
• Beiliang Du
• Hantao Zhang
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 563)

## Abstract

We shall refer to a diagonal Latin square which is orthogonal to its (3, 2, 1)-conjugate, and the latter is also a diagonal Latin square, as a (3, 2, 1)-conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS with a missing sub-square. The main purpose of this paper is two-fold. First of all, we show that for any positive integers n ≥ 1, a CODLS of order υ with a missing subsquare of order n exists if υ ≥ 13n/4 + 93 and υ - n is even. Secondly, we show that for 2 ≤ n ≤ 6, a CODLS of order υ with a missing subsquare of order n exists if and only if υ ≥ 3n + 2 and υ - n is even, with one possible exception.

Stein Tral Dene

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### References

1. [1]
F.E. Bennett, B. Du, and H. Zhang, Existence of conjugate orthogonal diagonal Latin squares, J. Combin. Designs, 5(1997), 449–461.
2. [2]
F.E. Bennett, B. Du, and H. Zhang, Existence of self-orthogonal diagonal Latin squares with a missing subsquare, Discrete Math., to appear.Google Scholar
3. [3]
F.E. Bennett, L. Wu, and L. Zhu, Conjugate orthogonal Latin squares with equal-sized holes, Ann. Discrete Math., 34(1987), 6580.Google Scholar
4. [4]
F.E. Bennett, L. Wu and L. Zhu, Further results on incomplete(3,2,1)-conjugate orthogonal idempotent Latin squares, Discrete Math., 84(1990), 1–14.
5. [5]
F.E. Bennett and L. Zhu, Incomplete conjugate orthogonal idempotent Latin squares, Discrete Math., 65(1987), 5–21.
6. [6]
F.E. Bennett and L. Zhu, Conjugate-orthogonal Latin squares and related structures, J. Dinitz & D. Stinson (Editors), Contemporary design theory: A collection of surveys, Wiley, New York, 1992, pp. 41–96.Google Scholar
7. [7]
C.J. Colbourn and J.H. Dinitz, The CRC handbook of combinatorial designs, CRC Press, Inc., Boca Raton, 1996.
8. [8]
J. Niles and A.D. Keedwell, Latin squares and their applications, Academic Press, New York and London, 1974.Google Scholar
9. [9]
J. Doyen and R.M. Wilson, Embeddings of Steiner triple systems, Discrete Math., 5(1973), 229–239.
10. [10]
B. Du, Self-orthogonal diagonal Latin square with missing sub-square, JCMCC, 37(2001), 193–203.
11. [11]
M. Greig, Designs from projective planes and PBD bases, J. Combin. Designs 7 (1999), 341–374.
12. [12]
M.J. Pelling and D.G. Rogers, Stein quasigroups I: combinatorial aspects, Bull. Austral. Math. Soc., 18(1978), 221–236.
13. [13]
S.K. Stein, On the foundations of quasigroups, Trans. Amer. Math. Soc., 85(1957), 228–256.
14. [14]
R.M. Wilson, Concerning the number of mutually orthogonal Latin squares, Discrete Math., 9(1974), 181–198.
15. [15]
R.M. Wilson, Constructions and uses of pairwise balanced designs, Math. Centre Tracts, 55(1974), 18–41.Google Scholar
16. [16]
H. Zhang and F.E. Bennett, Existence of some(3, 2, 1)-HCOLS and(3, 2, 1)-ICOILS, JCMCC, 27(1998), 53–64.

## Copyright information

© Kluwer Academic Publishers 2003

## Authors and Affiliations

• Frank E. Bennett
• 1
• Beiliang Du
• 2
• Hantao Zhang
• 3
1. 1.Department of MathematicsMount Saint Vincent UniversityHalifaxCanada
2. 2.Department of MathematicsSuzhou UniversitySuzhouP.R.China
3. 3.Computer Science DepartmentThe University of IowaIowa CityUSA