Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

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.     

On partitions of {1,…,2m + 1}\{k} into differences d,…,d + m − 1: Extended Langford sequences of large defect

It is shown that for m = 2d ? 1, 2d, 2d + 1, and d ≥ 1, the set {1, 2,…, 2m + 2}, ? {2,k} can be partitioned into differences d,d + 1,…,d + m ? 1 whenever (m,k) ≡ (0,0), (1,d + 1), (2, 1), (3,d) (mod (4,2)) and (d,m,k) ≠ (1,1,3), (2,3,7) (where (x,y) ≡ (u,ν) mod (m,n) iff x ≡ u (mod m) and y ≡ ν (mod n)). It is also shown that if m ≥ 2d ? 1 and m ? [2d + 2, 8d ? 5], then the set {1, 2, …, 2m + 1} ? {k} can be partitioned into differences d,d + 1,…,d + m ? 1 whenever (m,k) ≡ (0, 1), (1,d), (2,0), (3,d + 1) mod (4,2). Finally, for d = 4 we obtain a complete result for when {1,…,2m + 1} ? {k} can be partitioned into differences 4,5,…,m + 3. © 2004 Wiley Periodicals, 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     

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.     

3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, C_λ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: X → Z_m such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ^?1(c)∣ c ∈ Z_m differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5^a 13^b 17^c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

Large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ have been studied by Schellenberg, Chen, Lindner and Stinson. These large sets have applications in cryptography in the construction of perfect threshold schemes. It is known that such large sets can exist only for n ≡ 0 (mod 3) and do exist for n = 6 and for all n = 3^k, k ≥ 1. In this paper, we present new recursive constructions and use them to show that such large sets exist for all odd n ≡ 0 (mod 3) and for even n = 24m, where m odd ≥ 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 285–296, 2001     

Large sets of resolvable MTS and DTS of orders pn + 2

An LRMTS(v) [resp., LRDTS(v)] is a large set consisting of v − 2 [resp., 3(v − 2)] disjoint resolvable Mendelsohn (resp., directed) triple systems of order v. In this article, we give a method to construct LRMTS(pⁿ + 2) and LRDTS(pⁿ + 2), where pⁿ is a prime power and pⁿ ≡ 1 (mod 6). Using the method and a recursive construction v → 3v, some unknown LRMTS(v) and LRDTS(v) are obtained such as for v = 69, 123, 141, 159, and 3^km, where k ≥ 1, m ϵ {7, 13, 37, 55, 57, 61, 65, 67}. © 1996 John Wiley & Sons, Inc.     

Existence of directed GDDs with block size five

In this article, we construct directed group divisible designs (DGDDs) with block size five, group-type hⁿ, and index unity. The necessary conditions for the existence of such a DGDD are n ≥ 5, (n − 1)h ≡ 0 (mod 2) and n(n − 1)h² ≡ 0 (mod 10). It is shown that these necessary conditions are also sufficient, except possibly for n = 15 where h ≡ 1 or 5 (mod 6) and h ≢ 0 (mod 5), or (n, h) = (15, 9). © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 389–402, 1998     

Hooked extended Langford sequences of small and large defects

It is shown that for m = 2d +5, 2d+6, 2d+7 and d ≥ 1, the set {1, …, 2m + 1} ? {k} can be partitioned into differences d, d + 1, …, d + m ? 1 whenever (m, k) ≡ (0, 1), (1, d), (2, 0), (3, d+1) (mod (4, 2)) and 1 ≤ k ≤ 2m+1. It is also shown that for m = 2d + 5, 2d + 6, 2d + 7, and d ≥ 1, the set {1, …, 2m + 2} ? {k, 2m + 1} can be partitioned into differences d, d + 1, … …, d + m ? 1 whenever (m, k) ≡ (0, 0), (1, d + 1), (2, 1), (3, d) (mod (4, 2)) and k ≥ m + 2. These partitions are used to show that if m ≥ 8d + 3, then the set {1, … …, 2m+2}?{k, 2m+1} can be partitioned into differences d, d+1, …, d+m?1 whenever (m, k) ≡ (0, 0), (1, d+1), (2, 1), (3, d) (mod (4, 2)). A list of values m, d that are open for the existence of these partitions (which are equivalent to the existence of Langford and hooked Langford sequences) is given in the conclusion.     

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.     

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     

Extended skolem sequences

A k-extended Skolem sequence of order n is an integer sequence (s₁, s₂,…, s_2n+1) in which s_k = 0 and for each j ? {1,…,n}, there exists a unique i ? {1,…, 2n} such that s_i = s_i+j = j. We show that such a sequence exists if and only if either 1) k is odd and n ≡ 0 or 1 (mod 4) or (2) k is even and n ≡ 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. © 1995 John Wiley & Sons, Inc.     

Mutually disjoint designs and new 5‐designs derived from groups and codes

The article gives constructions of disjoint 5‐designs obtained from permutation groups and extremal self‐dual codes. Several new simple 5‐designs are found with parameters that were left open in the table of 5‐designs given in (G. B. Khosrovshahi and R. Laue, t‐Designs with t⩾3, in “Handbook of Combinatorial Designs”, 2nd edn, C. J. Colbourn and J. H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 79–101), namely, 5−(v, k, λ) designs with (v, k, λ)=(18, 8, 2m) (m=6, 9), (19, 9, 7m) (m=6, 9), (24, 9, 6m) (m=3, 4, 5), (25, 9, 30), (25, 10, 24m) (m=4, 5), (26, 10, 126), (30, 12, 440), (32, 6, 3m) (m=2, 3, 4), (33, 7, 84), and (36, 12, 45n) for 2⩽n⩽17. These results imply that a simple 5−(v, k, λ) design with (v, k)=(24, 9), (25, 9), (26, 10), (32, 6), or (33, 7) exists for all admissible values of λ. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 305–317, 2010     

Asymptotic Determination of the Last Packing Number of Quadruples

A 3-(n,4,1) packing design consists of an n-element set X and a collection of 4-element subsets of X, called blocks, such that every 3-element subset of X is contained in at most one block. The packing number of quadruples d(3,4,n) denotes the number of blocks in a maximum 3-(n,4,1) packing design, which is also the maximum number A(n,4,4) of codewords in a code of length n, constant weight 4, and minimum Hamming distance 4. In this paper the last packing number A(n,4,4) for n≡ 5(mod 6) is shown to be equal to Johnson bound with 21 undecided values n=6k+5, k∈{m: m is odd , 3≤ m≤ 35, m≠ 17,21}∪ {45,47,75,77,79,159}. AMS Classification:05B40, 94B25     

Path coverings with paths

In this paper, we give two new proofs of a result of Heinrich, Langdeau and Verrall that provide necessary and sufficient conditions for the existence of a set S of 3‐paths in K_n having the property that each 2‐path in K_n lies in exactly one path in S. These are then used to consider the case n ≡ 3 (mod 4) when no such exact covering is possible, and to solve the problem of covering (k−1)‐paths with k‐paths for all k ≥ 3. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 156–167, 2001     

Positive integers expressible as a sum of three squares in essentially only one way

The set S consisting of those positive integers n which are uniquely expressible in the form n = a² + b² + c², $\text{a ≧ b ≧ c ≧ 0}$, is considered. Since n ∈ S if and only if 4n ∈ S, we may restrict attention to those n not divisible by 4. Classical formulas and the theorem that there are only finitely many imaginary quadratic fields with given class number imply that there are only finitely many n ∈ S with n = 0 (mod 4). More specifically, from the existing knowledge of all the imaginary quadratic fields with odd discriminant and class number 1 or 2 it is readily deduced that there are precisely twelve positive integers n such that n ∈ S and n ≡ 3 (mod 8). To determine those n ∈ S such that n ≡ 1, 2, 5, 6 (mod 8) requires the determination of the imaginary quadratic fields with even discriminant and class number 1, 2, or 4. While the latter information is known empirically, it has not been proved that the known list of 33 such fields is complete. If it is complete, then our arguments show that there are exactly 21 positive integers n such that n ∈ S and n ≡ 1, 2, 5, 6 (mod 8).     

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.     

Transverse quadruple systems with five holes

Transverse Steiner quadruple systems with five holes are either of type g⁵ or g⁴u¹. We concentrate on the systems of type g⁴u¹ and settle existence except when g ≡ u ≡ 2 (mod 4) and all except 40 parameter situations when g ≡ u + 2 ≡ 0 (mod 4). The question of existence for transverse quadruple systems of type g⁴u¹ with index λ > 1 is completely solved for all λ ≥ 13 and λ ∈ {4, 6, 8, 9, 10, 12}. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 315–340, 2007     

Pairwise balanced designs whose block size set contains seven and thirteen

In this paper, we investigate the PBD‐closure of sets K with {7,13} ? K ? {7,13,19,25,31,37,43}. In particular, we show that ν ≡ 1 mod 6, ν ≥ 98689 implies ν ? B({7,13}). As an intermediate result, many new 13‐GDDs of type 13^q and resolvable BIBD with block size 6 or 12 are also constructed. Furthermore, we show some elements to be not essential in a Wilson basis for the PBD‐closed set {ν: ν ≡ 1 mod 6, ν ≥ 7}. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 283–314, 2007