Characterization of complete exterior sets of conics

Let be a set of exterior points of a nondegenerate conic inPG(2,q) with the property that the line joining any 2 points in misses the conic. Ifq1 (mod 4) then consists of the exterior points on a passant, ifq3 (mod 4) then other examples exist (at least forq=7, 11, ..., 31).Support from the Dutch organization for scientific Research (NWO) is gratefully acknowledged     

Construction of orthogonal starters and corollaries

A method is proposed for constructing a system of (v–1)/2 pairwise disjoint orthogonal starters of order v for v6k+17 (mod 12)pn²+n+1/t such that the number 3 is one of the primitive roots of the Galois field of prime order p (k is prime, k 2, and n and t are positive integers). The starters occurring in this system satisfy certain additional conditions. The construction of a series of combinatorial structures, including some not previously known, is a consequence of this result.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 654–662, May, 1992.     

ExplicitN-polynomials of 2-power degree over finite fields,I

For an odd prime powerq the infinite field GF(q ² )= _n0 GF (q ²ⁿ ) is explicitly presented by a sequence (f _n)₁ ofN-polynomials. This means that, for a suitably chosen initial polynomialf ₁, the defining polynomialsf _nGF(q)[x] of degrees2 ⁿ are constructed by iteration of the transformation of variablexx+1/x and have linearly independent roots over GF(q). In addition, the sequences are trace-compatible in the sense that the relative traces map the corresponding roots onto each other. In this first paper the caseq1 (mod 4) is considered and the caseq3 (mod 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].     

Some (17q, 17, 2) and (25q, 25, 3) BIBD Constructions

We give the explicit construction of a regular (17q, 17, 2)-BIBD for any prime power q 17 (mod 32) such that 2 is not a 4th power in GF(q) and the explicit construction of a regular (25q, 25, 3)-BIBD for any prime power q 25 (mod 48) such that and +3 are non-squares in GF(q).     

Largek-arcs in inversive planes of odd order

An infinite family of largek-arcs in the inversive plane over a finite field GF(q), withq 1 (mod 3),q71 orq {17,23, 27,29,41,47,49,53,59} is constructed.Research supported by G.N.S.A.G.A. of C.N.R., project Applicazioni della matematica per la tecnologia e la società, subproject Calcolo simbolico.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

C k -factorization of complete bipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of aC _k-factorization ofK _m,n is (i)m = n 0 (mod 2), (ii)k 0 (mod 2),k 4 and (iii) 2n 0 (modk) with precisely one exception, namely m =n = k = 6.     

New Hadamard matrices and conference matrices obtained via Mathon's construction

We give a formulation, via (1, –1) matrices, of Mathon's construction for conference matrices and derive a new family of conference matrices of order 59^2t+1 + 1,t 0. This family produces a new conference matrix of order 3646 and a new Hadamard matrix of order 7292. In addition we construct new families of Hadamard matrices of orders 69^2t+1 + 2, 109^2t+1 + 2, 8499 ^t ,t 0;q ²(q + 3) + 2 whereq 3 (mod 4) is a prime power and 1/2(q + 5) is the order of a skew-Hadamard matrix); (q + 1)q ²9 ^t ,t 0 (whereq 7 (mod 8) is a prime power and 1/2(q + 1) is the order of an Hadamard matrix). We also give new constructions for Hadamard matrices of order 49 ^t 0 and (q + 1)q ² (whereq 3 (mod 4) is a prime power).This work was supported by grants from ARGS and ACRB.Dedicated to the memory of our esteemed friend Ernst Straus.     

Existence of 1-Rotational <Emphasis Type="Italic">k</Emphasis>-Cycle Systems of the Complete Graph

We explicitly solve the existence problem for 1-rotational k-cycle systems of the complete graph K_v with v1 or k (mod 2k). For v1 (mod 2k) we have existence if and only if k is an odd composite number. For any odd k and vk (mod 2k), (except k3 and v15, 21 (mod 24)) a 1-rotational k-cycle system of K_v exists.Final version received: June 18, 2003     

Set systems with no union of cardinality 0 modulom

Letq be a prime power. It is shown that for any hypergraph = {F ₁,...,F _d(q–1)+1} whose maximal degree isd, there exists Ø ₀ , such that 0 (modq).     

Zerlegung total gerichteter Graphen in Kreise

The following statements are valid:The complete directed graph ¯K_n, n1 (mod 2p), is decomposable into directed 2p-cycles.The complete directed bipartite graph ¯K_m,n is decomposable into 2p-cycles if p is a divisor of m and np.If p is a prime, then this condition is necessary, too.The complete directed graph ¯K_n, n12, is decomposable into 6-cycles if and only if 6     

On dominant operators

This paper is devoted to the study of dominant operators with an emphasis on their spectral properties. In particular the equation (T–)f() x (T a dominant or hyponormal operator on the Hilbert space ,x andf a function from the open setU to ) is investigated in an effort to discover necessary and/or sufficient conditions for the analyticity off.Supported in part by the National Science Foundation.     

Odd Systems of Vectors and Related Lattices

We consider uniform odd systems, i.e. sets of vectors of constant odd norm with odd inner product, and the lattice L(V) linearly generated by a uniform odd system V of odd norm 2t+1. If uu p (mod 4) for all u V, one has v² p (mod 4) if v² is odd and v² 0 (mod 4) if v² is even, for any vector v L(V). The vectors of even norm form a double even sublattice L₀(V) of L(V), i.e. is an even lattice. The closure of V, i.e. all vectors of L(V) of norm 2t+1, are minimal vectors of L(V) for t=1, and they are almost always minimal for t=2. For such t, the convex hull of vectors of the closure of V is an L-polytope of L₀V and the contact polytope of L(V). As an example, we consider closed uniform odd systems of norm 5 spanning equiangular lines.     

Über die Quadratwurzel-Schranke für Quadratische-Rest-Codes

The minimal distanced of any QR-Code of lengthn 3mod4 over a prime fieldGF (p) with p3 mod4 satisfies the improved square root bound d(3d-2)4(n–1).

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Helmut Karzel zum 60. Geburtstag gewidmet     

Weakly union-free maximum packings

Frankl and Füredi [11] established that the largest number of 3-subsets of ann-set, for which no four distinct setsA,B,D satisfyAB=CD, is at most . Chee, Colbourn, and Ling [6] established that this upper bound is met with few exceptions whenn0, 1 (mod 3). In this paper, it is established that the upper bound is also met with few exceptions whenn2 (mod 3).The research was supported in part by the US Army Research Office under Grant DAAG55-98-1-0272.     

P 3-factorization of complete multipartite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of aP ₃-factorization ofK _m ⁿ is (i)mn 0(mod 3) and (ii) (m – 1)n 0(mod 4).     

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

Semiregular Large Sets

We develop the notion of t-homogeneous, G-semiregular large sets of t-designs, show that there are infinitely many 3-homogeneous PSL(2, q)-semiregular large sets when q 3 mod 4, two sporadic 3-homogeneous AL(1,32)-semiregular large sets, and no other interesting t-homogeneous G-semiregular large sets for t 3.     

On a Criterion for Catalan's Conjecture

We give a new proof of a theorem of P. Mihailescu which states that the equation x ^p – y ^q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p ^q p (mod q ²) and q ^p q (mod p ²) hold.     

Quasi-symmetric designs and the Smith Normal Form

We obtain necessary conditions for the existence of a 2 – (, k, ) design, for which the block intersection sizes s ₁, s ₂, ..., s _n satisfy s ₁ s ₂ ... s _n s (mod p ^e ),where p is a prime and the exponent e is odd. These conditions are obtained from restriction on the Smith Normal Form of the incidence matrix of the design. We also obtain restrictions on the action of the automorphism group of a 2 – (, k, ) design on points and on blocks.