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     

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.     

Existence of 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 (denoted by LS (2ⁿ4¹)) were first studied by Schellenberg and Stinson and motivated by 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 any n ? {12, 36, 48, 144} ∪ {m > 6 : m ≡ 6,30 (mod 36)}. In this paper, we show that an LS (2^{12k + 6}4¹) exists for any k ≠ 2. So, the existence of LS (2ⁿ4¹) is almost solved with five possible exceptions n ∈ {12, 30, 36, 48, 144}. This solution is based on the known existence results of S (3, 4, v)s by Hanani and special S (3, {4, 6}, 6m)s by Mills. Partitionable H (q, 2, 3, 3) frames also play an important role together with a special known LS (2¹⁸4¹) with a subdesign LS (2⁶4¹). © 2004 Wiley Periodicals, Inc.     

Some new HSOLSSOMs of types hn and 1m u1

In this paper, two new direct construction methods are given for holey self‐orthogonal Latin squares with a symmetric orthogonal mate (HSOLSSOMs). Some new HSOLSSOMs using already known methods are also given. The known existence results for HSOLSSOMs of types 1^m u¹ and hⁿ are improved; for type 1^m u¹ there remain just four possible exceptions with u odd and 3 ≤ u ≤ 15; for type hⁿ, there are just two possible exceptions remaining, for (h, n) = (6, 12) and (6, 18). As a byproduct, the known existence results for three holey mutually orthogonal Latin squares (3 HMOLS) are also improved. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 435–444, 2001     

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.     

Latin squares with restricted transversals

We prove that for all odd m ≥ 3 there exists a latin square of order 3 m that contains an ( m ? 1 ) × m latin subrectangle consisting of entries not in any transversal. We prove that for all even n ≥ 10 there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders n ≥ 11 . Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order n ? 9 . In particular, we count the number of species of each order that possess an orthogonal mate. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:124‐141, 2012     

A generalization of Fermat's little theorem

Fermat's Little Theorem states that if p is a prime number and gcd (x,p) = 1, then x^p?1 ≡ 1 (modp) If the requirement that gcd (x,p) = 1 is dropped, we can say x^p ≡ x(modp)for any integer x. Euler generalized Fermat's Theorem in the following way: if gcd (x,n) = 1 then x^φ(n) ≡ 1(modn), where φ is the Euler phi-function. It is clear that Euler's result cannot be extended to all integers x in the same way Fermat's Theorem can; that is, the congruence x^φ(n)+1 ≡ x(modn)is not always valid. In this note we show exactly when the congruence x^φ(n)+1 ≡ x(modn) is valid.     

On ε‐biased generators in NC0

Cryan and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272–284) recently considered the question of whether there can be a pseudorandom generator in NC⁰, that is, a pseudorandom generator that maps n‐bit strings to m‐bit strings such that every bit of the output depends on a constant number k of bits of the seed. They show that for k = 3, if m ≥ 4n + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a linear test, that is, there is a subset of bits of the output whose XOR has a noticeable bias. They leave the question open for k ≥ 4. In fact, they ask whether every NC⁰ generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC⁰ generator can sample an ε‐biased space with negligible ε? We give a generator for k = 5 that maps n bits into cn bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2. For large values of k, we construct generators that map n bits to bits such that every XOR of outputs has bias . We also present a polynomial‐time distinguisher for k = 4,m ≥ 24n having constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once m ≥ Ω(2^kn^?k/2?). Finally, we consider a variant of the problem where each of the output bits is a degree k polynomial in the inputs. We show there exists a degree k = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2^?Ω(n) and which maps n bits to Ω(n²) bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Balanced incomplete block designs with block size 8

The necessary conditions for the existence of a balanced incomplete block design on υ ≥ k points, with index λ and block size k, are that: For k = 8, these conditions are known to be sufficient when λ = 1, with 38 possible exceptions, the largest of which is υ = 3,753. For these 38 values of υ, we show (υ, 8, λ ) BIBDs exist whenever λ > 1 for all but five possible values of υ, the largest of which is υ = 1,177, and these five υ's are the only values for which more than one value of λ is open. For λ>1, we show the necessary conditions are sufficient with the definite exception of two further values of υ, and the possible exception of 7 further values of υ, the largest of which is υ=589. In particular, we show the necessary conditions are sufficient for all λ> 5 and for λ = 4 when υ ≠ 22. We also look at (8, λ) GDDs of type 7^m. Our grouplet divisible design construction is also refined, and we construct and exploit α ‐ frames in constructing several other BIBDs. In addition, we give a PBD basis result for {n: n ≡ 0, 1; mod 8, n ≥ 8}, and construct a few new TDs with index > 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 233–268, 2001     

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.     

Lucas's criterion for the primality of numbers of the form N = h2n−1

The following theorem is proved. Let N = h2ⁿ-1, where n ≥ 2, h is odd, 1 <-h < 2ⁿ, and suppose that v is a positive integer, v ≥ 3,α is a root of the equation $$(v^2 - 4,N) = 1,\left( {\frac{{v - 2}}{N}} \right) = 1,\left( {\frac{{v + 2}}{N}} \right) = - 1$$ . Then for N to be prime, it is necessary and sufficient that s_n?2≡0(modN), where S_k+1=S _k ² ? 2 (k = 0, 1...), s_o=a^h+ a^?h. For given N, an algorithm is described for the construction of the smallest v satisfying the conditions of this theorem.     

Sur le développement en fractions continues a quotients partiels impairs

Let (B _n ) _n be the sequence of denominators of convergents, given by the continued fraction expansion with odd partial quotients of an irrational number. For integersm>2 andk, 0k<m, we give (almost everywhere) the density of the set of integersn such thatB _n is congruent tok modm. This result comes from an ergodic system.

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Recherche partiellement subventionnée par l'URA n^o225, CNRS, Marseille.     

Resolvable Mendelsohn triple systems with equal sized holes

An HMTS of type {n₁, n₂, ⋖, n_h} is a directed graph which can be decomposed into 3-circuits. If the 3-circuits can be partitioned into parallel classes, then the HMTS is called an RHMTS. In this article it is shown that the RHMTSs of type m^h exist when mh &equiv 0 (mod 3) and (m, h) &ne (1, 6), with the possible exception of h = 6 and , where M₁₇ = {m|m is divisible by a prime less than 17}. The existence of Mendelsohn frames, which is closely related to RHMTS, is also considered in this article. It is proved that a Mendelsohn frame of type t^u exists if and only if u ≥ 4 and t(u - 1) ≡ 0(mod 3) with 2 possible exceptions. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:329–340, 1997     

The spectrum of additive BIB designs

In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sⁿ, ts^m, λt (ts^m ? 1) (s^n‐m ? 1)/[2(s^m ? 1)]) for any positive integer λ, such that sⁿ (sⁿ ? 1) λ ≡ 0 (mod s^m (s^m ? 1) and for any positive integer t with 2 ≤ t ≤ s^n‐m, where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v = 4k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007     

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.     

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.     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

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     

Coloring uniform hypergraphs with few colors

Let m(r, k) denote the minimum number of edges in an r‐uniform hypergraph that is not k‐colorable. We give a new lower bound on m(r, k) for fixed k and large r. Namely, we prove that if k ≥ 2ⁿ, then m(r, k) ≥ ?(k)k^r(r/ln r)^n/(n+1). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

On a conjecture of J. Pelikán

There exist infinitely many finite sequences (a₁, …, a_n) (a_i {0, 1}) such that Φ_{i = 1}^{n − k} a_ia_{i + k} is odd for each k = 0, 1, …, n − 1.