Designs 2002 pp 23-45 | Cite as

# Conjugate Orthogonal Diagonal Latin Squares with Missing Subsquares

## 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* ≥ 13*n*/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* ≥ 3*n* + 2 and *υ* - *n* is even, with one possible exception.

## Keywords

Positive Integer Disjoint Subset Parallel Class Multiplication Table Steiner Triple System## Preview

Unable to display preview. Download preview PDF.

## References

- [1]F.E. Bennett, B. Du, and H. Zhang,
*Existence of conjugate orthogonal diagonal Latin squares*, J. Combin. Designs,**5**(1997), 449–461.CrossRefMATHMathSciNetGoogle Scholar - [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]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]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.CrossRefMATHMathSciNetGoogle Scholar - [5]F.E. Bennett and L. Zhu,
*Incomplete conjugate orthogonal idempotent Latin squares*, Discrete Math.,**65**(1987), 5–21.CrossRefMATHMathSciNetGoogle Scholar - [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]C.J. Colbourn and J.H. Dinitz,
*The CRC handbook of combinatorial designs*, CRC Press, Inc., Boca Raton, 1996.CrossRefMATHGoogle Scholar - [8]J. Niles and A.D. Keedwell,
*Latin squares and their applications*, Academic Press, New York and London, 1974.Google Scholar - [9]J. Doyen and R.M. Wilson,
*Embeddings of Steiner triple systems*, Discrete Math.,**5**(1973), 229–239.CrossRefMATHMathSciNetGoogle Scholar - [10]B. Du,
*Self-orthogonal diagonal Latin square with missing sub-square*, JCMCC,**37**(2001), 193–203.MATHGoogle Scholar - [11]M. Greig,
*Designs from projective planes and PBD bases*, J. Combin. Designs**7**(1999), 341–374.CrossRefMATHMathSciNetGoogle Scholar - [12]M.J. Pelling and D.G. Rogers,
*Stein quasigroups I: combinatorial aspects*, Bull. Austral. Math. Soc.,**18**(1978), 221–236.CrossRefMATHMathSciNetGoogle Scholar - [13]S.K. Stein,
*On the foundations of quasigroups*, Trans. Amer. Math. Soc.,**85**(1957), 228–256.CrossRefMATHMathSciNetGoogle Scholar - [14]R.M. Wilson,
*Concerning the number of mutually orthogonal Latin squares*, Discrete Math.,**9**(1974), 181–198.CrossRefMATHMathSciNetGoogle Scholar - [15]R.M. Wilson,
*Constructions and uses of pairwise balanced designs*, Math. Centre Tracts,**55**(1974), 18–41.Google Scholar - [16]H. Zhang and F.E. Bennett,
*Existence of some*(3, 2, 1)-*HCOLS and*(3, 2, 1)-*ICOILS*, JCMCC,**27**(1998), 53–64.MATHMathSciNetGoogle Scholar