Proper S(t,K,v)'s for t ≥ 3,v ≤ 16, |K| > 1 and Their Extensions

We characterize the proper t-wise balanced designs t-(v,K,1) for t ≥ 3, λ = 1 and v ≤ 16 with at least two block sizes. While we do not examine extensions of S(3,4,16)'s, we do determine all other possible extensions of S(3,K,v)'s for v ≤ 16. One very interesting extension is an S(4, {5,6}, 17) design.©1995 John Wiley & Sons, Inc.     

On the extendability of steiner t-designs

Necessary and sufficient conditions for the extendability of residual designs of Steiner systems S(t,t + 1,v) are studied. In particular, it is shown that a residual design with respect to a single point is uniquely extendable, and the extendability of a residual design with respect to a pair of points is equivalent to a bipartition of the block graph of a related design. © 1993 John Wiley & Sons, Inc.     

A. translation of J. X. Lu's “an existence theory for resolvable balanced incomplete block designs”

In 1984, J. X. Lu proved the following statement. Given any k and λ, there exists a constant c(k, λ) such that an RB[v,k,λ] exists for all v > c(k,λ) satisfying the usual necessary conditions. Lu's paper was written in Chinese and, unfortunately, its accessibility is limited. We have translated Lu's paper into English and have also given a new interpretation of his constructions for resolvable block designs. © 1995 John Wiley & Sons, Inc.     

Note on large sets of infinite steiner systems

Large sets of Steiner systems S(t,k,n) exist for all finite t and k with t < k and all infinite n. The vector space analogues exist over a field F for all finite t and k with t < k provided that either v or F is infinite, and n ? 2k ? t + 1. This inequality is best possible. © 1995 John Wiley & Sons, Inc.     

The bigraphical t-wise balanced designs of index two

In this article we study the group S_m × S_n acting on the mn ordered pairs and classify all t-wise balanced designs of index 2 that have such an automorphism group. © 1995 John Wiley & Sons, Inc.     

New constructions for covering designs

A(v, k, t) covering design, or covering, is a family of k-subnets, called blocks, chosen from a v-set, such that each t-subnet is contained in at least one of the blocks. The number of blocks is the covering's size, and the minimum size of such a covering is denoted by C(v,k,t). This paper gives three new methods for constructing good coverings; a greedy algorithm similar to Conway and Sloane's algorithm for lexicographic codes [6], and two methods that synthesize new coverings from preexisting ones. Using these new methods, together with results in the literature, we build tables of upper bounds on C(v,k,t) for v ? 32, k ? 16, and t ? 8. © 1995 John Wiley & Sons, Inc.     

Smoothness versus skewness in block designs with an application to tournaments

The quadratic residue designs over GF(7) and GF(11) are shown to be the only quasismooth skew (Hadamard) designs, thus setting a conjecture of Herzog and Reid concerning the existence of so-called “nearly triply regular” tournaments. © 1995 John Wiley & Sons., Inc.     

On a class of small 2-designs over gf(q)

A 2 ? (v,k,λ;q) design is a pair (V, B) of a v-dimensional vector space V over GF(q) and a collection B of k-dimensional subspaces of V such that each 2-dimensional subspace of V is contained in exactly λ members of B. Assuming transitivity of their automorphism groups on the nonzero vectors of V, we give a classification of nontrivial such designs for v = 7, q = 2,3 with small λ, together with the nonexistence proof of those designs for v ? 6. © 1995 John Wiley & Sons, Inc.     

New coverings of t-sets with (t + 1)-sets

The minimum number of k-subsets out of a v-set such that each t-set is contained in at least one k-set is denoted by C(v, k, t). In this article, a computer search for finding good such covering designs, leading to new upper bounds on C(v, k, t), is considered. The search is facilitated by predetermining automorphisms of desired covering designs. A stochastic heuristic search (embedded in the general framework of tabu search) is then used to find appropriate sets of orbits. A table of upper bounds on C(v, t + 1, t) for v 28 and t 8 is given, and the new covering designs are listed. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 217–226, 1999     

Embeddings of pure mendelsohn triple systems and pure directed triple systems

It is proved in this article that the necessary and sufficient conditions for the embedding of a λ-fold pure Mendelsohn triple system of order v in λ-fold pure Mendelsohn triple of order u are λu(u ? 1) ≡ 0 (mod 3) and u ? 2v + 1. Similar results for the embeddings of pure directed triple systems are also obtained. © 1995 John Wiley & Sons, Inc.     

The existence of doubly near resolvable (v,3,2)-BIBDs

The existence of doubly near resolvable (v,2,1)-BIBDs was established by Mullin and Wallis in 1975. In this article, we determine the spectrum of a second class of doubly near resolvable balanced incomplete block designs. We prove the existence of DNR(v,3,2)-BIBDs for v ≡ 1 (mod 3), v ≥ 10 and v ? {34,70,85,88,115,124,133,142}. The main construction is a frame construction, and similar constructions can be used to prove the existence of doubly resolvable (v,3,2)-BIBDs and a class of Kirkman squares with block size 3, KS₃(v,2,4). © 1994 John Wiley & Sons, Inc.     

A few more large sets of t-designs

We construct several new large sets of t-designs that are invariant under Frobenius groups, and discuss their consequences. These large sets give rise to further new large sets by means of known recursive constructions including an infinite family of large sets of 3 − (v, 4, λ) designs. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 293–308, 1998     

Total dominating functions in trees: Minimality and convexity

A total dominating function (TDF) of a graph G = (V, E) is a function f: V ← [0, 1] such that for each v ? V, Σ_u?N(v) f(u) ≥ 1 (where N(v) denotes the set of neighbors of vertex v). Convex combinations of TDFs are also TDFs. However, convex combinations of minimal TDFs (i.e., MTDFs) are not necessarily minimal. In this paper we discuss the existence in trees of a universal MTDF (i.e., an MTDF whose convex combinations with any other MTDF are also minimal). © 1995 John Wiley & Sons, Inc.     

Embeddings of Sλ(2, 4, v)

It is shown that the necessary condition for a given S_λ(2,4,u) to be embedded in some S_λ(2,4,v) as a subdesign, namely v ≥ 3u + 1, is also sufficient for the case of λ = 6. Combining this with the previously known results gives the same sufficiency for any positive integer λ. © 1994 John Wiley & Sons, Inc.     

Some results on (v,{5,w*})-pbds

In this article, we construct pairwise balanced designs (PBDs) on v points having blocks of size five, except for one block of size w ? {17,21,25,29,33}. A necessary condition for the existence of such a PBD is v ? 4w + 1 and (1) v ≡ 1 or 5 (mod 20) for w = 21, 25; (2) v ≡ 9 or 17 (mod 20) for w = 17,29; (3) v ≡ 13 (mod 20) for w = 33. We show that these necessary conditions are sufficient with at most 25 possible exceptions of (v,w). We also show that a BIBD B(5, 1; w) can be embedded in some B(5, 1; v) whenever v ≡ w ≡ 1 or 5 (mod 20) and v ? 5w ? 4, except possibly for (v, w) = (425, 65). © 1995 John Wiley & Sons, Inc.     

Coloring BIBDs with block size 4

It is shown that for each integer m ≥ 1 there exists a lower bound, v_m, with the property that for all v ≥ v_m with v ≡ 1, 4 (mod 12) there exists an m-chromatic S(2, 4, v) design. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 403–409, 1998     

A class of Steiner 4-wise balanced designs derived from preparata codes

The minimum weight codewords in the Preparata code of length n = 4^m are utilized for the construction of an infinite family of Steiner S(4, {5, 6}, 4^m + 1) designs for any m ≥ 2. © 1996 John Wiley & Sons, Inc.     

Some new bibds with k = 6 and λ = 1

(v, 6,1) BIBDs are given for several new values of v. This reduces to 55 the number of v values for which existence of a (v,6,1) BIBD is in doubt. A new resolvable (565,5,1) BIBD is also given. © 1995 John Wiley & Sons, Inc.     

Overlarge sets of disjoint Steiner quadruple systems

In this article, we construct overlarge sets of disjoint S(3, 4, 3ⁿ − 1) and overlarge sets of disjoint S(3, 4, 3ⁿ + 1) for all n ≥ 2. Up to now, the only known infinite sequence of overlarge sets of disjoint S(3, 4, v) were the overlarge sets of disjoint S(3, 4, 2ⁿ) obtained from the oval conics of desarguesian projective planes of order 2ⁿ. © 1999 John Wiley & Sons, Inc. J Combin Design 7: 311–315, 1999     

An asymptotic approach for a semi‐Markovian inventory model of type (s,S)

In this study, we constructed a stochastic process (X(t)) that expresses a semi‐Markovian inventory model of type (s, S) and it is shown that this process is ergodic under some weak conditions. Moreover, we obtained exact and asymptotic expressions for the n^th order moments (n = 1,2,3, … ) of ergodic distribution of the process X(t), as S ? s → ∞ . Finally, we tested how close the obtained approximation formulas are to the exact expressions. Copyright © 2012 John Wiley & Sons, Ltd.