Partial steiner triple systems with equal-sized holes (original) (raw)

Group divisible designs with three groups and block size four

Discrete Mathematics, 2007

We present new constructions and results on GDDs with three groups and block size four and also obtain new GDDs with two groups of size nine. We say a GDD with three groups is even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups is always even, always odd, or always mixed. We give new necessary conditions for these families of GDDs and prove that they are sufficient for these three types and for all group sizes except for the minimal case of mixed designs for group size 5t(t>1)5t(t>1). In particular, we prove that mixed GDDs allow a maximum difference between indices. We apply the constructions given to show that the necessary conditions are sufficient for all GDDs with three groups and group sizes two, three, and five, and also for group size four with two possible exceptions, a GDD(4,3,4;5,9)GDD(4,3,4;5,9) and a GDD(4,3,4;7,12)GDD(4,3,4;7,12).

Covering triples by quadruples: An asymptotic solution

Journal of Combinatorial Theory, Series A, 1986

be the minimum number of four-element subsets (called blocks) of an n-element set, X, such that each three-element subset of X is contained in at least one block. Let L(3,4, n) = rn/4rn -1/3rn -2/2111. Schoenheim has shown that C(3,4, n) 2 L(3,4, n). The construction of Steiner quadruple systems of all orders n ~2 or 4 (mod 6) by Hanani (Canad. J. Math. 12 (1960), 145-157) can be used to show that C(3,4, n)= L(3,4, n) for all n=2, 3,4 or 5 (mod 6) and all n = 1 (mod 12). The case n = 7 (mod 12) is made more difficult by the fact that C(3,4,7) = L(3,4,7) + 1 and until recently no other value for C(3,4, n) with n=7 (mod 12) was known. In 1980 Mills showed by construction that C(3,4,499) = L(3,4,499).

Resolvable group divisible designs with block size 3

Discrete Mathematics, 1989

Let v be a non negative integer, let λ be a positive integer, and let K and M be sets of positive integers. A group divisible design, denoted by GD[K, λ, M, v], is a triple (X, Г, β) where X is a set of points, is a partition of X, and β is a class of subsets of X with the following properties. (Members of Г are called groups and members of β are called blocks.) 1. The cardinality of X is v.2. The cardinality of each group is a member of M.3. The cardinality of each block is a member of K.4. Every 2-subset {x, y} of X such that x and y belong to distinct groups is contained inprecisely λ blocks.5. Every 2-subset {x, y} of X such that x and y belong to the same group is contained in noblock.A group divisible design is resolvable if there exists a partition Π = {P1, P2,…} of β such that each part Pi is itself a partition of X. In this paper we investigate the existence of resolvable group divisible designs with K = {3}, M a singleton set, and all λ. The case where M = {1} has been solved by Ray-Chaudhuri and Wilson for λ = 1, and by Hanani for all λ > 1. The case where M is a singleton set, and λ = 1 has recently been investigated by Rees and Stinson. We give some small improvements to Rees and Stinson's results, and give new results for the cases where λ > 1. We also investigate a class of designs, introduced by Hanani, which we call frame resolvable group divisible designs and prove necessary and sufficient conditions for their existence.

Odd and even group divisible designs with two groups and block size four

Discrete mathematics, 2004

We show the necessary conditions are su cient for the existence of GDD(n; 2; 4; 1, 2) with two groups and block size four in which every block intersects each group exactly twice (even GDD's) or in which every block intersects each group in one or three points (odd GDD's). We give a construction for near 3-resolvable triple systems TS(n; 3; 6) for every n ¿ 4, and these are used to provide constructions for several families of GDDs.

Optimal partitions for triples

Journal of Combinatorial Theory, Series A, 1992

The best known method to obtain constant weight codes with distance 4 is the partitioning method. To apply this method one has to partition sets of n-tuples into disjoint constant codes of weight w and distance 4, such that the number of codes will be minimal and the codes will be as large as possible. In this paper we consider the case of w = 3. For n I 0, 1,2, 3 (mod 6) the optimal partition is derived from disjoint Steiner triple systems. We give optimal partitions for all order n = 3k + 1, k=l or 5 (mod6). For orders n=3k+2, k=l or 5 (mod6), k>l, or kz3 (mod 12) we present constructions which get the maximal number, 3k, of disjoint optimal codes. We give a construction which gets a partition of order n = qk + i, i=l,2, q=l or S(mod6), q>l, k=3(mod6), with qk disjoint optimal codes, if a partition of order k + i with k optimal codes exists, and a set of q pairwise disjoint Steiner triple systems, or order q + 2, with some special property exists. For these q's, we also prove that for n =9q+2 there exists a partition with 9q + 1 codes.

GBRDs with block size 3 over odd order groups and groups of orders divisible by 2 but not 4

Australas. J Comb., 2012

Well-known necessary conditions for the existence of a generalized Bhaskar Rao design, GBRD(v, 3, λ; G) with v ≥ 4 are: (i ) λ ≡ 0 (mod |G|), (ii ) λ(v -1) ≡ 0 (mod 2), (iii ) λv(v -1) ≡ 0 (mod 3), (iv ) if |G| ≡ 0 (mod 2) then λv(v -1) ≡ 0 (mod 8). In this paper we show that these conditions are sufficient whenever (i ) the group G has odd order or (ii ) the order is of the form 2q for q = 3 m or q an odd number which is not a multiple of 3.