Existence of six incomplete MOLS

Six mutually orthogonal Latin squares of order v (briefly 6 MOLS(v» missing a subdesign of 6 MOLS(n) are denoted by 6 IMOLS(v, n). The necessary condition for their existence is v ;;::: 7n. For small v and n where 1 v 1000 and 0 :::; n :::; 50, the existence has been investigated by Col bourn and Dinitz, who give a table of possible exceptions in the CRC Handbook of Combinatorial In this paper, we study the existence of 6 IMOLS(v, n) when v and n are beyond the above range. we show that 6 IMOLS(v, n) always exist when v 1000 and 0 n 50. Second, we extend the table to list the possible exceptions when 51 :::; n :::; 97. For n ;;::: 98, we are able to show that 6 IMOLS(v, n) exist whenever v 8n + 139. We improve this in several ways, including that for all n ;:::: 23 when n is a prime power, and for all n 781, the necessary condition v;;::: 7n is also sufficient. Some results on 4 IMOLS(v, n) are also mentioned. 1. Introdu.ction Let S be a set and H = {S 1> square having hole set H is an lSI properties: ... , Sn} be a set of subsets of S. A holey Latin lSI array L, indexed by S, satisfying the following (l) every cell of L either contains a symbol of S or is empty, every symbol of S occurs at most once in any row or column of L, (3) the subarrays indexed by Si x Si are empty for 1 :::; i:::; n (these subarrays are referred to as holes), (4) symbol s E S occurs in row or column t if and only if (s, t) (SxS)\ul~ i ~(SixS0. * on leave from: Department of Mathematics, Suzhou University, Suzhou, 215006, China Aust.-alasian Journal of Combinatot~ics .:!lJ1995L pp.175-191 The order of L is lSI. If the holes are ..,.,'i .... ~I·'0'" UJ."'IVUll., the Latin square denoted by Two holey Latin squares and M on if their superposition the notation k sl, Sn) to denote where any two of them ortnlOgcmal. If H = we obtain k If H {S 1 }, we simply write k 2, further the notation by 51)' Let value of k for which k It is well known that k MOLS(s) are equivalent to a TD(k + 2, for eX'imlPle. k n) prl1l1"'.1 "",'C"Ir to v) + 2, MOLS and k IMOLS played an lrYlT,r."";''lnt role in the construction of various kinds of combinatorial In [11], Horton started to look at the existence k IMOLS. Simple counting shows the following. Lemma 1.1 If there exist k .,V'" HJ,J\ n), then v (k + 1 )n. When k 2, the existence has been completely solved in [10]. Theorem 1.2 [10] For any (v, n) = 1). n 1, IMOLS(v, n) exist and only if v 3n, except For k = 3, the existence was solved in when n 154. Du [7], has lowered the bound and listed lO9 of (v, nY possible Abel, Colbourn and Yin have further reduced the list to 24 possible ex(~eDtions. which we state as follows. Theorem 1.3 [1] For any except (v, n) (6, 1) and n 1, there ~~r'=~1h~ ~ the 24 :n possible exceptions 10 Table 1.1 24 Dm,sllJ,le e:xcevtions of 176 n) if and only if v 4n of n) shown in Table 1.1.