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A directed covering design, DC(v, k, λ), is a (v, k, 2λ) covering design in which the blocks are regarded as ordered k-tuples and in which each ordered pair of elements occurs in at least λ blocks. Let DE(v, k, λ) denote the minimum number of blocks in a DC(v, k, λ). In this paper the values of the function DC(v, 5, λ) are determined for all even integers v ≥ 5 and λ odd.

In this paper we improve and generalize the results of two earlier papers of Skillicorn. The maximum (minimum) number of blocks in a directed (v; 4 ;� ) packing (covering) designs are completely determined. c 2001 Elsevier Science B.V. All rights reserved.

In this paper it is shown that the MGD[4,1, m, mn] exist when (m − 1)(n − 1) ≡ 1 (mod 3) and n, m ⩾ 4 with some possible exceptions.

Let v and k be positive integers. A (v, k, 1)-packing design is an ordered pair (V, B) where V is a v-set and B is a collection of k-subsets of V (called blocks) such that every 2-subset of V occurs in at most one block of B. The packing problem is mainly to determine the packing number P(k, v), that is, the maximum number of blocks in such a packing design. It is well known that P(k, v) ≤ ⌊v⌊(v − 1)/(k − 1)⌋/k⌋ = J(k, v) where ⌊×⌋ denotes the greatest integer y such that y ≤ x. A (v, k, 1)-packing design having J(k, v) blocks is said to be optimal. In this article, we develop some general constructions to obtain optimal packing designs. As an application, we show that P(5, v) = J(5, v) if v ≡ 7, 11 or 15 (mod 20), with the exception of v ∈ {11, 15} and the possible exception of v ∈ {27, 47, 51, 67, 87, 135, 187, 231, 251, 291}. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 245–260, 1998

LetV be a finite set of order υ. A (υ, κ, λ) covering design of index λ and block size κ is a collection of κ-element subsets, called blocks, such that every 2-subset ofV occurs in at least λ blocks. The covering problem is to determine the minimum number of blocks, α(υ, κ, λ), in a covering design. It is well known that\(\alpha \left( {\upsilon ,\kappa ,\lambda } \right) \geqslant \left\lceil {\tfrac{v}{\kappa }\left\lceil {\tfrac{{v - 1}}{{\kappa - 1}}\lambda } \right\rceil } \right\rceil = \phi \left( {\upsilon ,\kappa ,\lambda } \right)\), where [x] is the smallest integer satisfyingx≤[X]. It is shown here that α(υ, 5, 5)=ϕ(υ, 5, 5) for all positive integers υ≥5 with the possible exception of υ=24, 28, 56, 104, 124, 144, 164, 184.

A ( ν , κ , λ ) packing design of order ν, block size κ, and index λ is a collection of κ-element subsets, called blocks, of a ν-set V such that every 2-subset of V occurs in at most λ blocks. The packing problem is to determine the maximum number of blocks in a packing design. Packing with λ =2 is called bipacking. In this paper we solve the bipacking problem in the case κ =5 and ν ≡13 ( mod 20).

A (v, k, λ) packing design of order v, block size k, and index λ is a collection of k-element subsets, called blocks, of a v-set, V, such that every 2-subset of V occurs in at most λ blocks. The packing problem is to determine the maximum number of blocks in a packing design. In this paper we provide a powerful technique for constructing designs and solve the packing problem with k = 5, λ = 2, and all odd numbers v, v ≢ 13 (mod 20), with the possible exception of v = 19, 27, 137, 139, 147.

A (v, k. λ) covering design of order v, block size k, and index λ is a collection of k-element subsets, called blocks, of a set V such that every 2-subset of V occurs in at least λ blocks. The covering problem is to determine the minimum number of blocks, α(v, k, λ), in a covering design. It is well known that , where [χ] is the smallest integer satisfying χ ≤ χ. It is shown here that α (v, 5, λ) = ϕ(v, 5, λ) + ϵ where λ ≡ 0 (mod 4) and e= 1 if λ (v−1)≡ 0(mod 4) and λv (v−1)/4 ≡ −1 (mod 5) and e= 0 otherwise With the possible exception of (v,λ) = (28, 4). © 1993 John Wiley & Sons, Inc.

A (v, к, λ) packing design of order v, block size к and index λ is a collection of к-element subsets, called blocks, of a v-set V such that every 2-subset of V occurs in at most λ blocks. The packing problem is to determine the maximum number of blocks in a packing design. The only previous work on the packing problem with к=6 concerns itself with the cases where the maximum packing design is in fact a balanced incomplete block design. In this paper we solve the packing problem with к=6 and λ=5 and all positive integers v with the possible exceptions of v=41, 47, 53, 59, 62, 71.

In this paper it is shown that for λ even, (5,λ)-MGDDs of type mn exists whenever the known necessary conditions are satisfied, with a finite number of exceptions.

A (υ, κ, λ) packing design of order υ, block size κ, and index λ is a collection of κ-element subsets, called blocks of a set V such that every 2-subset of V occurs in at most λ blocks. The packing problem is to determine the maximum number of blocks in a packing design. In this paper we solve the packing problem with κ = 5, λ = 8, 12, 16, and all positive integers υ with the possible exceptions of (υ, λ) = (19, 16) (22, 16) (24, 16) (27, 16) (28, 12).

An (υ, κ, λ) packing design of order υ, block size κ, and index λ is a collection of κ-element subsets, called blocks, of a set V such that every 2-subsets of V occurs in at most λ blocks. The packing problem is to determine the maximum number of blocks in a packing design. In this paper we provide a powerful technique for constructing designs and solve the packing problem in the case κ = 5, λ = 2, and υ is even.

Let V be a finite set of order m × n , and assume that the points of V are arranged in an array of size m × n . A modified group divisible design is a group divisible design with the difference that each 2-subset { x , y } of V such that x and y are neither in the same row nor in the same column occurs λ times. In this paper we apply modified group divisible designs to construct covering designs, packing designs, and group divisible designs with block size 5.

In this article we look at pair covering designs with a block size of 5 and v≡0(mod4). The number of blocks in a minimum covering design is known as the covering number C(v,5,2). For v⩽24, these values are known, and all but v=8 exceed the Schönheim bound, L(v,5,2)=⌈v/5⌈(v-1)/4⌉⌉. However, for all v⩾28 with v≡0(mod4), it seems probable that C(v,5,2)=L(v,5,2). We establish this for all but 17 possible exceptional values lying in the range 40⩽v⩽280.