Existence of (q, 7, 1) difference families with q a prime power

The existence of a (q,k, 1) difference family in GF(q) has been completely solved for k = 3,4,5,6. For k = 7 only partial results have been given. In this article, we continue the investigation and use Weil's theorem on character sums to show that the necessary condition for the existence of a (q,7,1) difference family in GF(q), i.e. q ≡ 1; (mod 42) is also sufficient except for q = 43 and possibly except for q = 127, q = 211, q = 31⁶ and primes q∈ [261239791, 1.236597 × 10¹³] such that in GF(q). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 126–138, 2002; DOI 10.1002/jcd.998     

Improving two theorems of bose on difference families

In [2] R. C. Bose gives a sufficient condition for the existence of a (q, 5, 1) difference family in (GF(q), +)—where q ≡ 1 mod 20 is a prime power — with the property that every base block is a coset of the 5th roots of unity. Similarly he gives a sufficient condition for the existence of a (q, 4, 1) difference family in (GF(q, +)—where q ≡ 1 mod 12 is a prime power — with the property that every base block is the union of a coset of the 3rd roots of unity with zero. In this article we replace the mentioned sufficient conditions with necessary and sufficient ones. As a consequence, we obtain new infinite classes of simple difference families and hence new Steiner 2-designs with block sizes 4 and 5. In particular, we get a (p^4α, 5, 1)-DF for any odd prime p ≡ 2, 3 (mod 5), and a (p^2α, 4, 1)-DF for any odd prime p ≡ 2 (mod 3). © 1995 John Wiley & Sons, Inc.     

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.     

Large sets of 3-designs from psl(2, q), with block sizes 4 and 5

We determine the distribution of 3?(q + 1,k,λ) designs, with k ? {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ? 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3?(q + 1,k,λ) designs, with k ? {4,5} and q ≡ 3 (mod 4). © 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.     

Existence results for KS3(v; 2, 4)s

A Kirkman square with index λ, latinicity μ, block size k and v points, KS_k(v; μ, λ), is a t × t array (t = λ(v − 1)/μ(k − 1)) defined on a v-set V such that (1) each point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (v, k, λ)-BIBD. The existence question for KS₂(v; μ, λ) has been completely selttled. We are interested in the next case k = 3. The case k = 3 and μ = λ = 1 appears to be quite difficult, although some existence results are available. For λ > 1 and μ ⩾ 1, the problem is more tractable. In this paper, we prove the existence of KS₃(v; 2, 4) for v ≡ 3 (mod 12), v ≡ 6 (mod 60) and v ≡ 9 (mod 96).     

Infinite families of 2- and 3-designs with parameters v = p + 1, k = (p − 1)/2i + 1, where p odd prime, 2e T (p − 1), e ≥ 2, 1 ≤ i ≤ e

Let p be an odd prime number such that p − 1 = 2^em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2ⁱ + 1 and λ = ((p − 1)/2ⁱ+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 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     

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.     

Some new BIBDS with λ = 1 and 6 ≤ k ≤ 10

This article is in two main parts. The first gives some (q,k, 1) difference families with q a prime power and 7 ≤ k ≤ 9; it also gives some GD(k, 1, k,kq)s which are extendable to resolvable (kq,k, 1) BIBDs for k E {6,8,10} and q a prime power equal to 1 mod 2(k − 1). The second uses some of these plus several recursive constructions to obtain some new (v,k,, 1) BIBDs with 7 ≤ k ≤ 9 and some new (v,8,1) resolvable BIBDs. © 1996 John Wiley & Sons, Inc.     

Existence results for near resolvable designs

A near resolvable design, NRB(v, k), is a balanced incomplete block design whose block set can be partitioned into v classes such that each class contains every point of the design but one, and each point is missing from exactly one class. The necessary conditions for the existence of near resolvable designs are v ≡ 1 mod k and λ = k ? 1. These necessary conditions have been shown to be sufficient for k ? {2,3,4} and almost always sufficient for k ? {5,6}. We are able to show that there exists an integer n₀(k) so that NRB(v,k) exist for all v > n₀(k) and v ≡ 1 mod k. Using some new direct constructions we show that there are many k for which it is easy to compute an explicit bound on n₀(k). These direct constructions also allow us to build previously unknown NRB(v,5) and NRB(v,6). © 1995 John Wiley & Sons, Inc.     

On simple radical difference families

For q a prime power and k odd (even), we define a (q,k,1) difference family to be radical if each base block is a coset of the kth roots of unity in the multiplicative group of GF(q) (the union of a coset of the (k ? 1)th roots of unity in the multiplicative group of GF(q) with zero). Such a family will be denoted by RDF. The main result on this subject is a theorem dated 1972 by R.M. Wilson; it is a sufficient condition for the existence of a (q,k, 1)-RDF for any k. We improve this result by replacing Wilson's condition with another sufficient but weaker condition, which is proved to be necessary at least for k ? 7. As a consequence, we get new difference families and hence new Steiner 2-designs. © 1995 John Wiley & Sons, Inc.     

Cyclic k‐cycle systems of order 2kn + k: A solution of the last open cases

We exhibit cyclic (K_v, C_k)‐designs with v > k, v ≡ k (mod 2k), for k an odd prime power but not a prime, and for k = 15. Such values were the only ones not to be analyzed yet, under the hypothesis v ≡ k (mod 2k). Our construction avails of Rosa sequences and approximates the Hamiltonian case (v = k), which is known to admit no cyclic design with the same values of k. As a particular consequence, we settle the existence question for cyclic (K_v, C_k)‐designs with k a prime power. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 299–310, 2004.     

Parametrization of Triangle Groups as Subgroups of PSL(2, q)

In this paper we are interested in triangle groups (j, k, l) where j = 2 and k = 3. The groups (j, k, l) can be considered as factor groups of the modular group PSL(2, Z) which has the presentation x, y : x² = y³ = 1. Since PSL(2,q) is a factor group of G^k,l,m if -1 is a quadratic residue in the finite field F_q, it is therefore worthwhile to look at (j, k, l) groups as subgroups of PSL(2, q) or PGL(2, q). Specifically, we shall find a condition in form of a polynomial for the existence of groups (2, 3, k) as subgroups of PSL(2, q) or PGL(2, q).Mathematics Subject Classification: Primary 20F05 Secondary 20G40.     

On schematic orthogonal arraysOA(q t,k, q,t)

Am × k matrixA, with entries from a set ofq 2 elements, is called an orthogonal arrayOA(m, k, q, t) (t 2) if eachm × t submatrix ofA contains all possible 1 ×t row vectors with the same frequency(m = q ^t ). We call the array schematic if the set of rows ofA forms an association scheme with the relations determined by the Hamming distance. In this paper we determine the schematic orthogonal arraysOA(q ^t ,k, q, t) with2t – 1 > k.     

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.     

On the Constant Metric Dimension of Generalized Petersen Graphs P（n, 4）

In this paper,we consider the family of generalized Petersen graphs P(n,4).We prove that the metric dimension of P(n,4) is 3 when n ≡ 0(mod 4),and is 4 when n = 4k + 3(k is even).For n ≡ 1,2(mod 4) and n = 4k + 3(k is odd),we prove that the metric dimension of P(n,4) is bounded above by 4.This shows that each graph of the family of generalized Petersen graphs P(n,4)has constant metric dimension.     

Optimal holey packings OHP4(2, 4, n ,3)'s

An optimal holey packing OHP_d(2, k, n, g) is equivalent to a maximal (g + 1)‐ary (n, k, d) constant weight code. In this paper, we provide some recursive constructions for OHP_d(2, k, n, g)'s and use them to investigate the existence of an OHP₄(2, 4, n, 3) for n ≡ 2, 3 (mod 4). Combining this with Wu's result ( 18 ), we prove that the necessary condition for the existence of an OHP₄(2, 4, n, 3), namely, n ≥ 5 is also sufficient, except for n ∈ {6, 7} and except possibly for n = 26. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 111–123, 2006     

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.     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.