On the decomposition of λK into regular tournaments of order 5

An RTD[5,λ; v] is a decomposition of the complete symmetric directed multigraph, denoted by λK, into regular tournaments of order 5. In this article we show that an RTD[5,λ; v] exists if and only if (v?1)λ ≡ 0 (mod 2) and v(v?1)λ ≡ 0 (mod 10), except for the impossible case (v,λ) = (15,1). Furthermore, we show that for each v ≡ 1,5 (mod 20), v ≠ 5, there exists a B[5,2; v] which is not RT₅-directable. © 1994 John Wiley & Sons, Inc.     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

On optimal (v, 5, 2, 1) optical orthogonal codes

The size of a (v, 5, 2, 1) optical orthogonal code (OOC) is shown to be at most equal to ${\lceil{\frac{v}{12}}\rceil}$ when v ≡ 11 (mod 132) or v ≡ 154 (mod 924), and at most equal to ${\lfloor{\frac{v}{12}}\rfloor}$ in all the other cases. Thus a (v, 5, 2, 1)-OOC is naturally said to be optimal when its size reaches the above bound. Many direct and recursive constructions for infinite classes of optimal (v, 5, 2, 1)-OOCs are presented giving, in particular, a very strong indication about the existence of an optimal (p, 5, 2, 1)-OOC for every prime p ≡ 1 (mod 12).     

Bounds and Constructions on (v, 4, 3, 2) Optical Orthogonal Codes

In this paper, we are concerned about optimal (v, 4, 3, 2)‐OOCs. A tight upper bound on the exact number of codewords of optimal (v, 4, 3, 2)‐OOCs and some direct and recursive constructions of optimal (v, 4, 3, 2)‐OOCs are given. As a result, the exact number of codewords of an optimal (v, 4, 3, 2)‐OOC is determined for some infinite series.     

Completing the spectrum for LGDD (mv)

In this article, we construct LGDD(m^v) for v ≡ 2 (mod 6) and m ≡ 0 (mod 6) by constructing the missing essential values 6^v, v = 8, 14, 26, 50. Thus an LGDD(m^v) exists if and only if v(v - 1)m² ≡ 0 (mod 6), (v - 1)m ≡ 0 (mod 2) and (m, v) ≠ (1, 7). © 1997 John Wiley & Sons, Inc.     

Quintessential PBDs and PBDs with prime power block sizes ≥ 8

In this paper, we look at the existence of (v K) pairwise balanced designs (PBDs) for a few sets K of prime powers ≥ 8 and also for a number of subsets K of {5, 6, 7, 8, 9}, which contain {5}. For K = {5, 7}, {5, 8}, {5, 7, 9}, we reduce the largest v for which a (v, K)‐PBD is unknown to 639, 812, and 179, respectively. When K is Q_≥8, the set of all prime powers ≥ 8, we find several new designs for 1,180 ≤ v ≤ 1,270, and reduce the largest unsolved case to 1,802. For K =Q_0,1,5(8), the set of prime powers ≥ 8 and ≡ 0, 1, or 5 (mod 8) we reduce the largest unknown case from 8,108 to 2,612. We also obtain slight improvements when K is one of {8, 9} or Q_0,1(8), the set of prime powers ≡ 0 or 1 (mod 8). © 2004 Wiley Periodicals, 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.     

Small Uniformly Resolvable Designs for Block Sizes 3 and 4

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k‐pc and for a given k the number of k‐pcs is denoted r_k. In this paper, we consider the case of block sizes 3 and 4 (both existent). We use v to denote the number of points, in this case the necessary conditions imply that v ≡ 0 (mod 12). We prove that all admissible URDs with v < 200 points exist, with the possible exceptions of 13 values of r₄ over all permissible v. We obtain a URD({3, 4}; 276) with r₄ = 9 by direct construction use it to and complete the construction of all URD({3, 4}; v) with r₄ = 9. We prove that all admissible URDs for v ≡ 36 (mod 144), v ≡ 0 (mod 60), v ≡ 36 (mod 108), and v ≡ 24 (mod 48) exist, with a few possible exceptions. Recently, the existence of URDs for all admissible parameter sets with v ≡ 0 (mod 48) was settled, this together with the latter result gives the existence all admissible URDs for v ≡ 0 (mod 24), with a few possible exceptions.     

A bound for wilson's theorem (I)

In 1975, Richard M. Wilson proved: Given any positive integers k ? 3 and λ, there exists a constant v⁰ = v⁰(k, λ) such that v ? B(k,λ) for every integer v ? v⁰ that satisfies λ(v ? 1) ≡ 0(mod k ? 1) and λv(v ? 1) ≡ 0[mod k(k ? 1)]. The proof given by Wilson does not provide an explicit value of v⁰. We try to find such a value v⁰(k, λ). In this article we consider the case λ = 1 and v ≡ 1[mod k(k ? 1)]. We show that: if k ? 3 and v = 1[mod k(k ? 1)] where v > k^kk5, then a B(v,k, 1) exists. © 1995 John Wiley & Sons, Inc.     

A bound for Wilson's theorem (III)

In this article we prove the following statement. For any positive integers k ≥ 3 and λ, let c(k, λ) = exp{exp{k;rcub;}. If λv(v − 1) ≡ 0 (mod k(k − 1)) and λ(v − 1) ≡ 0 (mod k − 1) and v > c(k, λ), then a B(v, k, λ) exists. © 1996 John Wiley & Sons, Inc.     

Covering pairs by quintuples with index λ  0 (mod 4)

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 $ \alpha \left({\nu,\kappa,\lambda } \right) \ge \left\lceil {\frac{\nu}{\kappa}\left\lceil {\frac{{\nu - 1}}{{\kappa - 1}}\lambda} \right\rceil} \right\rceil = \phi \left({\nu,\kappa,\lambda} \right) $, where [χ] is the smallest integer satisfying χ ≤ χ. It is shown here that α (v, 5, λ) = ?(v, 5, λ) + ? 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.     

On the 3BD‐closed set B3({4,5,6})

Let B₃(K) = {v:? an S(3,K,v)}. For K = {4} or {4,6}, B₃(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if v ∈ B₃ ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B₃{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ? B₃({4,5,6}) by Hanani and that B₃({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ? B₃({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B₃({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.     

The spectrum for large sets of disjoint mendelsohn triple systems with any index

The spectrum for LMTS(v,1) has been obtained by Kang and Lei (Bulletin of the ICA, 1993). In this article, firstly, we give the spectrum for LMTS(v,3). Furthermore, by the existence of LMTS(v,1) and LMTS(v,3), the spectrum for LMTS(v,λ) is completed, that is v ≡ 2 (mod λ), v ≥ λ + 2, if λ ? 0(mod 3) then v ? 2 (mod 3) and if λ = 1 then v ≠ 6. © 1994 John Wiley & Sons, Inc.     

Z-cyclic triplewhist tournaments when the number of players involves primes of the form 8u + 5

When the number of players, v, in a whist tournament, Wh(v), is ≡ 1 (mod 4) the only instances of a Z-cyclic triplewhist tournament, TWh(v), that appear in the literature are for v = 21,29,37. In this study we present Z-cyclic TWh(v) for all v ∈ T = {v = 8u + 5: v is prime, 3 ≤ u ≤ 249}. Additionally, we establish (1) for all v ∈ T there exists a Z-cyclic TWh(vⁿ) for all n ≥ 1, and (2) if v_i ∈ T, i = 1,…,n, there exists a Z-cyclic TWh(v… v) for all ?_i ≥ 1. It is believed that these are the first instances of infinite classes of Z-cyclic TWh(v), v ≡ 1 (mod 4). © 1994 John Wiley & Sons, Inc.     

Ordered tournaments and ordered triplewhist tournaments with the three person property

It is well known that an ordered tournament OWh(v) exists if and only if v ≡ 1 (mod 4), v ≥ 5. An ordered triplewhist tournament on v players is said to have the three person property if no two games in the tournament have three common players. We briefly denote such a design as a 3POTWh(v). In this article, we show that a 3POTWh(v) exists whenever v>17 and v ≡ 1 (mod 4) with few possible exceptions. We also show that an ordered whist tournament on v players with the three person property, denoted 3POWh(v), exists if and only if v ≡ 1 (mod 4), v ≥ 9. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 39–52, 2009     

Optimal (4up, 5, 1) optical orthogonal codes

By a (ν, k, 1)‐OOC we mean an optical orthogonal code. In this paper, it is proved that an optimal (4p, 5, 1)‐OOC exists for prime p ≡ 1 (mod 10), and that an optimal (4up, 5, 1)‐OOC exists for u = 2, 3 and prime p ≡ 11 (mod 20). These results are obtained by applying Weil's theorem. © 2004 Wiley Periodicals, Inc.     

4‐*GDDs(3n) and generalized Steiner systems GS(2, 4, v, 3)

Generalized Steiner systems GS(2, k, v, g) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g + 1 with minimum Hamming distance 2k ? 3, in which each codeword has length v and weight k. As to the existence of a GS(2, k, v, g), a lot of work has been done for k = 3, while not so much is known for k = 4. The notion k‐*GDD was first introduced and used to construct GS(2, 3, v, 6). In this paper, singular indirect product (SIP) construction for GDDs is modified to construct GS(2, 4, v, g) via 4‐*GDDs. Furthermore, it is proved that the necessary conditions for the existence of a 4‐*GDD(3ⁿ), namely, n ≡ 0, 1 (mod 4) and n ≥ 8 are also sufficient. The known results on the existence of a GS(2, 4, v, 3) are then extended. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 381–393, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10047     

A bound for Wilson's theorem (II)

In this article we prove the following theorem. For any k ≥ 3, let c(k, 1) = exp{exp{k^k2}}. If v(v − 1) ≡ 0 (mod k(k −1)) and v − 1 ≡ 0 (mod k−1) and v > c(k, 1), then a B(v,k, 1) exists. © 1996 John Wiley & Sons, Inc.     

Resolvable packings of Kv with K2 × Kc's

The study of resolvable packings of K_v with K_r × K_c's is motivated by the use of DNA library screening. We call such a packing a (v, K_r × K_c, 1)‐RP. As usual, a (v, K_r × K_c, 1)‐RP with the largest possible number of parallel classes (or, equivalently, the largest possible number of blocks) is called optimal. The resolvability implies v ≡ 0 (mod rc). Let ρ be the number of parallel classes of a (v, K_r × K_c, 1)‐RP. Then we have ρ ≤ ?(v‐1)/(r + c ? 2)?. In this article, we present a number of constructive methods to obtain optimal (v, K₂ × K_c, 1)‐RPs meeting the aforementioned bound and establish some existence results. In particular, we show that an optimal (v, K₂ × K₃, 1)‐RP meeting the bound exists if and only if v ≡ 0 (mod 6). © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 177–189, 2009     

Constructions of optimal packing designs

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