Elementary constructions of locally compact affine planes

In this paper we describe several elementary constructions of 4-, 8- and 16-dimensional locally compact affine planes. The new planes share many properties with the classical ones and are very easy to handle. Among the new planes we find translation planes, planes that are constructed by gluing together two halves of different translation planes, 4-dimensional shift planes, etc. We discuss various applications of our constructions, e.g. the construction of 8- and 16-dimensional affine planes with a point-transitive collineation group which are neither translation planes nor dual translation planes, the proof that a 2-dimensional affine plane that can be coordinatized by a linear ternary field with continuous ternary operation can be embedded in 4-, 8- and 16-dimensional planes, the construction of 4-dimensional non-classical planes that admit at the same time orthogonal and non-orthogonal polarities. We also consider which of our planes have tangent translation planes in all their points. In a final section we generalize the Knarr-Weigand criterion for topological ternary fields.This research was supported by a Feodor Lynen fellowship.     

Differentiation evens out zero spacings

If is a polynomial with all of its roots on the real line, then the roots of the derivative are more evenly spaced than the roots of . The same holds for a real entire function of order 1 with all its zeros on a line. In particular, we show that if is entire of order 1 and has sufficient regularity in its zero spacing, then under repeated differentiation the function approaches, after normalization, the cosine function. We also study polynomials with all their zeros on a circle, and we find a close analogy between the two situations. This sheds light on the spacing between zeros of the Riemann zeta-function and its connection to random matrix polynomials.

     

Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.     

On self-dual codes over {\mathbb{F}}_5

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over for lengths [22, 26, 28, 32–40]. In particular, we prove that there is no [22, 11, 9] self-dual code over , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18, 9, 7] codes over up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over and at least 40 new inequivalent optimal self-dual [22, 11, 8] codes.      

On circulant self-dual codes over small fields

We construct self-dual codes over small fields with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.      

Symplectic semifield planes and -linear codes

There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and -linear Kerdock and Preparata codes on the other. These inter-relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non-isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of -linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2.

We also obtain large numbers of ``boring' affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes ``boring' in the sense that each has as small an automorphism group as possible.

The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.

     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

Equivariant deformations of LeBrun's self-dual metrics with torus action

We investigate -equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all -subgroups of the torus for which one can obtain -equivariant deformations that do not preserve the whole of the torus action. This gives many new self-dual metrics with -action which are not conformally isometric to LeBrun metrics. We also count the dimension of the moduli space of self-dual metrics with -action obtained in this way.

     

Constructions for Terraces and R‐Sequencings,Including a Proof That Bailey's Conjecture Holds for Abelian Groups

Every abelian group of even order with a noncyclic Sylow 2‐subgroup is known to be R‐sequenceable except possibly when the Sylow 2‐subgroup has order 8. We construct an R‐sequencing for many groups with elementary abelian Sylow 2‐subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2‐groups have terraces. For odd orders it is known that abelian groups are R‐sequenceable except possibly those with noncyclic Sylow 3‐subgroups. We show how the theory of narcissistic terraces can be exploited to find R‐sequencings for many such groups, including infinitely many groups with each possible of Sylow 3‐subgroup type of exponent at most 3¹² and all groups whose Sylow 3‐subgroups are of the form or .     

Tangentially transitive planes of order 16

Lorimer and Rahilly have given constructions for translation planes of order 16. We show that both of these planes have a subplane of order 2 and a group of collineations fixing pointwise which is transitive on the points of –. We show that all such planes of order 16 are isomorphic and that these planes have a number of other properties which are exceptional.     

Bruck nets with a transitive direction

We start the systematic investigation of the geometric properties and the collineation groups of Bruck nets N with a transitive direction (i.e. with a group G of central translations acting transitively on each line of a given parallel class P). After reviewing some basic properties of such nets (in particular, their connection to difference matrices), we shall consider the problem of what can be said if either N or G admits an interesting extension. Specifically, we shall handle the following four situations: (1) there is a second transitive direction; (2) N is a translation net (w.l.o.g. with translation group K containing G); (3) the dual of NP is a translation transversal design (w.l.o.g. with translation group K containing G); (4) N admits a transversal (and can then in fact be extended by adding a further parallel class). Our study of these problems will yield interesting generalizations of known concepts (e.g. that of a fixed-point-free group automorphism) and results (for affine and projective planes). We shall also see that a wide variety of seemingly unrelated results and constructions scattered in the literature are in fact closely related and should be viewed as part of a unified whole.To Helmut Salzmann on the occasion of his 60th birthdayThe results of this paper will form part of the first author's doctoral dissertation which is being written under the supervision of the second author.     

Group rings, <Emphasis Type="Italic">G</Emphasis>-codes and constructions of self-dual and formally self-dual codes

We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in Hurley (Int J Pure Appl Math 31(3):319–335, 2006) by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes.     

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

Dual Families of Interacting Particle Systems on Graphs

A simple condition for IPS (Interacting Particle Systems) with nearest neighbor interactions to be self-dual is given. It follows that any IPS with the contact transition and no spontaneous birth is self-dual. It is shown that families of IPS exist in which every IPS is dual to every other, and such that for every pair of IPS, one is a thinning of the other. Further, all such IPS have the same form for an equilibrium distribution when expressed in terms of survival probabilities. Convergence results from a wide class of initial infinite measures follow.     

Higher-order Carmichael numbers

We define a Carmichael number of order to be a composite integer such that th-power raising defines an endomorphism of every -algebra that can be generated as a -module by elements. We give a simple criterion to determine whether a number is a Carmichael number of order , and we give a heuristic argument (based on an argument of Erdos for the usual Carmichael numbers) that indicates that for every there should be infinitely many Carmichael numbers of order . The argument suggests a method for finding examples of higher-order Carmichael numbers; we use the method to provide examples of Carmichael numbers of order .

     

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.     

Affine Barbilian structures

W.Leissner has characterized, by geometric axioms, the affine Barbilian planes over a Z-ring (i.e, a ring with 1 such that ab=1 ba=1) [10].The aim of the present paper is to characterize correspondingly the affine Barbilian planes over an arbitrary ring with 1. First we shall deal with the translation Barbilian planes, which generalize Leissner's parallelodromic planes [11]. The paper concludes with a study of the kernel of the translation Barbilian plane.Here, the terms affine Barbilian structure and affine Barbilian plane are used in a more general sense than in [10] and [11]. Also, the definitions of translation and parallelodromy are slightly different from those in [10] and [11], insomuch that the invariance of the non-neighbour relation is not postulated any more, this being a consequence in a translation Barbilian plane.In H.J.Arnold's geometry of rings, for any two distinct points, there exists a smallest line incident with them [1]. This property, assumed only for the non-neighbour pairs of points, will replace the usual postulate that two non-neighbour points are incident with exactly one line. Thus, ideas of D.Barbilian [2] and H. J.Arnold [1] are combined with methods of affine ring-geometry due to J.Hjelmslev [5], [6], W.Klingenberg [7], [8], [9], H.Lüneburg [12], W.Benz [3], [4], W.Leissner [10], [11], and others. Many parts of the proofs in [10] and [11] could be used here almost unchanged, under relaxed assumptions.     

Eine Klasse von achtdimensionalen lokalkompakten Translationsebenen mit großen Scherungsgruppen

This paper is one of the final steps in a classification program to determine all eight-dimensional, locally compact translation planes having large collineation groups. Here, we describe all such planes whose collineation group contains a semidirect product ·N, whereN is an at least 3-dimensional normal subgroup consisting of shears with fixed axis, and is isomorphic to SO₃ ().     

Independent Domination in Cubic Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by , is the minimum cardinality of an independent dominating set. In this article, we show that if is a connected cubic graph of order n that does not have a subgraph isomorphic to K_{2, 3}, then . As a consequence of our main result, we deduce Reed's important result [Combin Probab Comput 5 (1996), 277–295] that if G is a cubic graph of order n, then , where denotes the domination number of G.