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 of (q,k, 1) difference families with q a prime power and k = 4, 5

The existence of a (q, k, 1) difference family in GF(q) has been completely solved for k = 3. For k = 4, 5 partial results have been given by Bose, Wilson, and Buratti. In this article, we continue the investigation and show that the necessary condition for the existence of a (q, k, 1) difference family in GF(q), i.e., q ≡ 1 (mod k(k − 1)) is also sufficient for k = 4, 5. For general k, Wilson's bound shows that a (q, k, 1) difference family in GF(q) exists whenever q ≡ 1 (mod k(k − 1)) and q > [k(k − 1)/2]^k(k−1). An improved bound on q is also presented. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 21–30, 1999     

On antipodal Euclidean tight (2e + 1)-designs

Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ ⁿ . For an integer t, a finite subset X of ℝ ⁿ given together with a weight function w is a Euclidean t-design if holds for any polynomial f(x) of deg(f)≤ t, where {S _i , 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X _i = X∩ S _i , and w:X→ ℝ_{> 0}. (The case of X⊂ S ⁿ⁻¹ with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.     

Existence of triple solutions of discrete (n,p) boundary value problems

We consider the following boundary value problem: −Δⁿy = F(k,y, Δy,…,Δⁿ⁻¹y), k ϵ Z[n − 1, N], Δⁱy(0) = 0, 0 ≤ i ≤ n − 2, Δ^py(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ p ≤ n − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions.     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

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.     

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.     

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.     

Mixed block designs

Let V_i (i = 1, 2) be a set of size v_i. Let D be a collection of ordered pairs (b₁, b₂) where b_i is a k_i-element subset of V_i. We say that D is a mixed t-design if there exist constants λ _(j,j2), (0 ≤ j_i ≤ k_i, j₁ + j₂ ≤ t) such that, for every choice of a j₁-element subset S₁ of V₁ and every choice of a j₂-element subset S₂ of V₂, there exist exactly λ_(j1,j2) ordered pairs (b₁, b₂) in D satisfying S₁ ⊆ b₁ and S₂ ⊆ b₂. In W. J. Martin [Designs in product association schemes, submitted for publication], Delsarte's theory of designs in association schemes is extended to products of Q-polynomial association schemes. Mixed t-designs arise as a particularly interesting case. These include symmetric designs with a distinguished block and α-resolvable balanced incomplete block designs as examples. The theory in the above-mentioned paper yields results on mixed t-designs analogous to those known for ordinary t-designs, such as the Ray-Chaudhuri/Wilson bound. For example, the analogue of Fisher's inequality gives |D| ≥ v₁ + v₂ − 1 for mixed 2-designs with Bose's condition on resolvable designs as a special case. Partial results are obtained toward a classification of those mixed 2-designs D with |D| = v₁ + v₂ − 1. The central result of this article is Theorem 3.1, an analogue of the Assmus–Mattson theorem which allows us to construct mixed (t + 1 − s)-designs from any t-design with s distinct block intersection numbers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:151–163, 1998     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let σ₂(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σ^k_{i = 1} a_i and σ₂(G) ≥ n + k − 1, then for any k vertices v₁, v₂,…, v_k in G, there exist vertex‐disjoint paths P₁, P₂,…, P_k such that |V(P_i)| = a_i and v_i is an endvertex of P_i for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all a_i ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000     

Meyniel's theorem for strongly (p,q) - Hamiltonian digraphs

We give the following theorem: Let D = (V, E) be a strongly (p + q + 1)-connected digraph with n ≥ p + q + 1 vertices, where p and q are nonnegative integers, p ≤ n - 2, n ≥ 2. Suppose that, for each four vertices u, v, w, z (not necessarily distinct) such that {u, v} ∩ {w, z} = Ø, (w, u) ? E, (v, z) ? E, we have id(u) + od(v) + od(w + id(z) ≥ 2 (n + p + q)) + 1. Then D is strongly (p, q)-Hamiltonian.     

Reducibility of polynomials a0(x) + a1 (x)y + a2 (x) y2 modulo p

Let f=a₀(x)+a₁(x)y+a₂(x)y² ? \Bbb Z[x,y]f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y] be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if p 3 c_m·H(f)^e_mp\ge c_m\cdot H(f)^{e_m} where H (f) is the height of f and e₁ = 4,e₂ = 6, e₃ = 6 [2/3]{2}\over{3} and e_m = 2 m for m S 4. We also show that the exponents e_m are best possible for m 1 3m\ne 3 if a plausible number theoretic conjecture is true.     

The intersection problem for m-cycle systems

Let I_m(v) denote the set of integers k for which a pair of m-cycle systems of K_v, exist, on the same vertex set, having k common cycles. Let J_m(v) = {0, 1, 2,…, t_v ?2, t_v} where t_v = v(v ? 1)/2m. In this article, if 2mn + x is an admissible order of an m-cycle system, we investigate when I_m(2mn + x) = J_m(2mn + x), for both m even and m odd. Results include J^m(2mn + 1) = I^m(2mn + 1) for all n > 1 if m is even, and for all n > 2 if n is odd. Moreover, the intersection problem for even cycle systems is completely solved for an equivalence class x (mod 2m) once it is solved for the smallest in that equivalence class and for K_2m+1. For odd cycle systems, results are similar, although generally the two smallest values in each equivalence class need to be solved. We also completely solve the intersection problem for m = 4, 6, 7, 8, and 9. (The cased m = 5 was done by C-M. K. Fu in 1987.) © 1993 John Wiley & Sons, Inc.     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

A 35-set of type (2, 5) in PG(2, 9)

A 35-set of type (2, 5) is constructed in the Desarguesian plane of order nine, which is the first example of a set of type (m, q + m) and (m + q)(q² − q + 1) points, with m = 2, in an odd square order plane.     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

A family of 2-weight codes related to BCH-codes

For every prime-power q and every pair of natural numbers m ≤ n′, we construct a q-ary linear code of length q^m (q^n′ − 1)(q^n′ − q^n′−m + 1)/(q − 1) and dimension 3n′, whose only nonzero weights are and . These code parameters and those of the corresponding family of strongly regular graphs are new in odd characteristic. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 391–396, 1997     

Construction of Z-cyclic triple whist tournaments

Let p = 2^kt + 1 be a prime where t>1 is an odd integer, k ≥ 2. Methods of constructing a Z-cyclic triple whist tournament TWh(p) are given. By such methods we construct a Z-cyclic TWh(p) for all primes p,p≡1(mod 4), 29 ≤ p ≤ 16097, except p = 257. Let p_i = 2t_i + 1,q = 2t₀ + 3 be primes where t_i;i = 0,1,…, n are odd > 1 and k_i are integers ≥2. We prove that if Z-cyclic TWh(p_i) and TWh(q + 1) exist then Z-cyclic TWh(∏ⁿ_{i = 1} p_i) and TWh(q∏ⁿ_{i = 1} p_i + 1) exist. © 1996 John Wiley & Sons, Inc.     

On the k-error linear complexity of sequences with period 2pn over GF(q)

In this paper, we first optimize the structure of the Wei–Xiao–Chen algorithm for the linear complexity of sequences over GF(q) with period N =  2p ⁿ , where p and q are odd primes, and q is a primitive root modulo p ². The second, an union cost is proposed, so that an efficient algorithm for computing the k-error linear complexity of a sequence with period 2p ⁿ over GF(q) is derived, where p and q are odd primes, and q is a primitive root modulo p ². The third, we give a validity of the proposed algorithm, and also prove that there exists an error sequence e ^N , where the Hamming weight of e ^N is not greater than k, such that the linear complexity of (s + e) ^N reaches the k-error linear complexity c. We also present a numerical example to illustrate the algorithm. Finally, we present the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.