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.     

Non-symmetric 2-designs modulo 2

Necessary conditions are obtained for the existence of a 2 – (v, k, ) design, for which the block intersection sizess ₁,s ₂, ...,s _n satisfys ₁ s ₂ ... s _n s (mod 2 ^e ), wheree is odd. These conditions are obtained by combining restrictions on the Smith Normal Form of the incidence matrix of the design with some well known properties of self-orthogonal binary codes with all weights divisible by 4.Research done at AT&T Bell Laboratories.     

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).     

Some Multiply Derived Translation Planes with SL(2,5) as an Inherited Collineation Group in the Translation Complement

Finite translation planes having a collineation group isomorphic to SL(2,5) occur in many investigations on minimal normal non-solvable subgroups of linear translation complements. In this paper, we are looking for multiply derived translation planes of the desarguesian plane which have an inherited linear collineation group isomorphic to SL(2,5). The Hall plane and some of the planes discovered by Prohaska [10], see also [1], are translation planes of this kind of order q ²;, provided that q is odd and either q ²; 1 mod 5 or q is a power of 5. In this paper the case q ² -1 mod 5 is considered and some examples are constructed under the further hypothesis that either q 2 mod 3, or q 1 mod 3 and q 1 mod 4, or q -1 mod 4, 3 q and q 3,5 or 6 mod 7. One might expect that examples exist for each odd prime power q. But this is not always true according to Theorem 2.     

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.     

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     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Incomplete idempotent Schröder quasigroups and related packing designs

Summary In this paper it is proved that, for any positive integern 2, 3 (mod 4),n 7, there exists an incomplete idempotent Schröder quasigroup with one hole of size two IISQ(n, 2) except forn = 10. It is also proved that for any positive integern 0, 1 (mod 4), there exists an idempotent Schröder quasigroup ISQ(n) except forn = 5 and 9. These results completely determine the spectrum of ISQ(n) and provide an application to the packing of a class of edge-coloured block designs.Research supported by NSERC grant A-5320.Research supported by NSFC grant 19231060-2.     

Resolvable Mendelsohn designs with block size 4

Summary Letv andK be positive integers. A (v, k, 1)-Mendelsohn design (briefly (v, k, 1)-MD) is a pair (X,B) whereX is av-set (ofpoints) andB is a collection of cyclically orderedk-subsets ofX (calledblocks) such that every ordered pair of points ofX are consecutive in exactly one block ofB. A necessary condition for the existence of a (v, k, 1)-MD isv(v–1) 0 (modk). If the blocks of a (v, k, 1)-MD can be partitioned into parallel classes each containingv/k blocks wherev ) (modk) or (v – 1)/k blocks wherev 1 (modk), then the design is calledresolvable and denoted briefly by (v, k, 1)-RMD. It is known that a (v, 3,1)-RMD exists if and only ifv 0 or 1 (mod 3) andv 6. In this paper, it is shown that the necessary condition for the existence of a (v, 4, 1)-RMD, namelyv 0 or 1 (mod 4), is also sufficient, except forv = 4 and possibly exceptingv = 12. These constructions are equivalent to a resolvable decomposition of the complete symmetric directed graphK _v ^* onv vertices into 4-circuits.Research supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-5320.     

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.     

On a Connection of Number Theory with Graph Theory

We assign to each positive integer n a digraph whose set of vertices is H = {0, 1, ..., n – 1} and for which there is a directed edge from a H to b H if a ² b (mod n). We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.     

On designs and formally self-dual codes

Binary formally self-dual (f.s.d.) even codes are the one type of divisible [2n, n] codes which need not be self-dual. We examine such codes in this paper. On occasion a f.s.d. even [2n, n] code can have a larger minimum distance than a [2n, n] self-dual code. We give many examples of interesting f.s.d even codes. We also obtain a strengthening of the Assmus-Mattson theore. IfC is a f.s.d. extremal code of lengthn2 (mol 8) [n 6 (mod 8)], then the words of a fixed weight inC C hold a 3-design [1-design]. Finally, we show that the extremal f.s.d. codes of lengths 10 and 18 are unique.The author thanks the University of Illinois at Chicago for their hospitality while this work was in progress.This work was supported in part by NSA Grant MDA 904-91-H-0003.     

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.     

Element neighbourhoods in twofold triple systems

The method of differences is used to establish that every 2-regular multigraph onv– 10,2 (mod 3) points occurs as the neighbourhood graph of an element in a twofold triple system of orderv, with two exceptions: C₂C₃and C₃C₃.Dedicated to Professor Hanfried Lenz on the occasion of his seventieth birthday     

Tables on fifth power residuacity

In this paper 2 ^p 1 (modq),q=10p+1,p 3 (mod 4),p andq prime, is expressed uniquely (except for changes in sign and interchange ofx, y) in the formq=w ²+25 (x ²+y ²)/2+125z ², 4wz=y ²–x ²–4xy, withw, x, y, z odd, forp<10⁵. For 10⁵<p<10⁶, allp such that 2 ^p 1 (mod 10p + 1),p 3 (mod 4),p and 10p + 1 prime, are listed.     

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     

Simple minimum coverings ofK n with copies ofK 4 −e

Summary AK ₄–e design of ordern is a pair (S, B), whereB is an edge-disjoint decomposition ofK _n (the complete undirected graph onn vertices) with vertex setS, into copies ofK ₄–e, the graph on four vertices with five edges. It is well-known [1] thatK ₄–e designs of ordern exist for alln 0 or 1 (mod 5),n 6, and that if (S, B) is aK ₄–e design of ordern then |B| =n(n – 1)/10.Asimple covering ofK _n with copies ofK ₄–e is a pair (S, C) whereS is the vertex set ofK _n andC is a collection of edge-disjoint copies ofK ₄–e which partitionE(K_n)P, for some . Asimple minimum covering ofK _n (SMCK _n) with copies ofK ₄–e is a simple covering whereP consists of as few edges as possible. The collection of edgesP is called thepadding. Thus aK ₄–e design of ordern isSMCK _n with empty padding.We show that forn 3 or 8 (mod 10),n 8, the padding ofSMCK _n consists of two edges and that forn 2, 4, 7 or 9 (mod 10),n 9, the padding consists of four edges. In each case, the padding may be any of the simple graphs with two or four edges respectively. The smaller cases need separate treatment:SMCK ₅ has four possible paddings of five edges each,SMCK ₄ has two possible paddings of four edges each andSMCK ₇ has eight possible paddings of four edges each.The recursive arguments depend on two essential ingredients. One is aK ₄–e design of ordern with ahole of sizek. This is a triple (S, H, B) whereB is an edge-disjoint collection of copies ofK ₄–e which partition the edge set ofK _n\K_k, whereS is the vertex set ofK _n, and is the vertex set ofK _k. The other essential is acommutative quasigroup with holes. Here we letX be a set of size 2n 6, and letX = {x ₁, x₂, ..., x_n} be a partition ofX into 2-element subsets, calledholes of size two. Then a commutative quasigroup with holesX is a commutative quasigroup (X, ) such that for each holex _i X, (x_i, ) is a subquasigroup. Such quasigroups exist for every even order 2n 6 [4].     

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.     

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     

Invariante und strenge Tests

Summary Using an idea of Kiefer [5] it is proved that an invariant and most stringent test as introduced by Schaafsma [7] exists, if the test problem is invariant under a solvable group of measurable transformations. Considering a test problem of Ajne [1] for uniformity of a circular distribution it is shown, however, that in this case is an invariant and most stringent size- test.