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.     

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.     

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.     

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.     

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.     

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     

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 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.     

On a property of symmetric designs of order n≡2 (mod 4)

Summary U. Ott, during his visit in Rome (spring 1985), by using the theory of even unimodular lattices, proved that a (v,k,) symmetric design of order n2 (mod 4) satisfies the congruence v ±1 (mod 8). He asked me the question whether this is a consequence of the Bruck-Ryser-Chowla's theorem or not. In this paper we prove that the answer to this question is affirmative. As a consequence of this, we have that the conjecture according to which the Bruck-Ryser-Chowla's theorem and the identity k²–v=n imply the existence of a (v,k,) symmetric design is still open.     

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).     

Rate of expansion of an inhomogeneous branching process of brownian particles

Summary Let X be the (B ⁰, {q _n (x)})-branching diffusion where B ⁰is the exp -subprocess of BM(R¹) and q _n (x) is the probability that a particle dying at x produces n offspring, q ₀ q ₁0. Put m(x) = nq _n (x). We assume q _n , n2, m and k are all continuous (but m is not necessarily bounded). If k(x)m(x)0 as ¦x¦, then we prove that R _t /t( ₂/2)^1/2, as t, a.s. and in mean (of any order) where R _t is the position of the rightmost particle at time t and ₀ is the largest eigenvalue of (1/2)d ²/dx ² + Q, Q(x) = k(x)(m(x)–1).This work was supported in part by a grant from the National Science Foundation # MCS-8201470.     

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     

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     

Nonvanishing of central Hecke L-values and rank of certain elliptic curves

Let D 7 mod 8 be a positive squarefree integer, and let h_D be the ideal class number of E_D= . Let d1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k0 there is a constant M=M(k), independent of the pair (D,D), such that if (–1)^k=sign (d), (2k+1,h_D)=1, and >(12/)d² (logd+M(k)), then the central L-value L(k+1, _{D, d} ^2k+1 >0. Furthermore, for k1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)^d has Mordell–Weil rank 0 (over its definition field) when >(12/)d² log d.     

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.     

There Exists a Simple Non Trivial t-design with an Arbitrarily Large Automorphism Group for Every t

In 1987, Teirlinckproved that if t and are two integers such that v t(mod(t + 1)!^(2t+1) and v t + 1 >0, then there exists a t - (v, t + 1, (t + 1)!^(2t+1)) design. We prove that if there exists a (t+1)-(v,k,)design and a t-(v-1,k-2, (k-t-1)/(v-k+1))design with t 2, then there exists a t-(v+1,k, (v-t+1)(v-t)/ (v-k+1)(k-t))design. Using this recursive construction, we prove that forany pair (t,n) of integers (t 2and n 0), there exists a simple non trivial t-(v,k,) design having an automorphism groupisomorphic to ⁿ ₂.     

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.     

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.     

A Solution of a Problem of A. E. Brouwer

Let q=p^e 1(mod 4) be a prime power, and let ^(q) be the Paley graph over the finite field . Denote by ^(q) the subgraph of ^(q) induced on the set of non-zero squares of . In this paper the full automorphism group of ^(q) is determined affirming the conjecture of Brouwer [Des. Codes Cryptograph. 21, 69–76 (2000)]. The proof combines spectral and Schur ring techniques.     

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].