Further results on incomplete (3,2,1)- conjugate orthogonal idempotent Latin squares

✍ Scribed by F.E. Bennett; Lisheng Wu; L. Zhu

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 878 KB
• Volume: 84
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(90)90267-l

✦ Synopsis

Let us denote by COILS(v) a (3,2, l)-conjugate orthogonal idempotent Latin square of order v, and by ICOILS(v, n) an incomplete COILS(v) missing a sub-COILS(n).

A necessary condition for the existence of an ICOILS(v,

and the necessary condition for its existence has recently been shown by the authors to be sufficient for all v 3 4 with the exception of v = 6 and the possible exception of v = 12. Two of the above authors have previously shown that for n > 1, an ICOILS(v, n) exists if v = 3n + 1 or v 2 8n + 42. Moreover, it was also shown that, for 2 c n L 6, an ICOILS(v, n) exists for all v 2 3n + 1 with some possible exceptions.

The main purpose of this paper is two-fold. First of all, for 2 c n =Z 6, we substantially reduce the number of possible exceptions and show that, in particular, the necessary condition is sufficient for n = 4, 5, and 6 except possibly when (v, n) = (30, 5). Secondly, we show that for n 2 1, an ICOILS(v, n) exists for all v 3 (13/4)n + 88, which gives a general bound much closer to the necessary condition.

* The author acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada under Grant A-5320.