Table Of ContentDESIGNS AND FINITE
GEOMETRIES
Edited by
Dieter Ju ngnickel
University ofA ugsburg
A Special Issue of
DESIGNS, CODES AND CRYPTOGRAPHY
An International Journal
Volume 8, No. 112 (1996)
KLUWER ACADEMIC PUBLISHERS
Boston / Dordrecht / London
DESIGNS, CODES
AND CRYPTOGRAPHY
An International Jo umal
Volume 8, No. 112, May 1996
Special Issue Dedicated to Han/ried Lenz
Guest Editor: Dieter lungnickel
Preface .............................................. Dieter Jungnickel 7
A Life's Work in Geometry: An Homage to Hanfried Lenz .................. .
· .................................... Dieter Jungnickel and Gunter Pickert 9
Impossibility of a Certain Cyclotomic Equation with Applications to Difference Sets
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. K. T. Arasu and Alexander Pott 23
On the Binary Codes of Steiner Triple Systems ............................ .
· ................. Alphonse Baartmans, Ivan Landjev, and Vladimir D. Tonchev 29
Orthogonal Partitions in Designed Experiments ................... R. A. Bailey 45
Regulus-free Spreads of PG(3, q) ................. R. D. Baker and G. L. Ebert 79
Designs, Codes and Crypts-A Puzzle Altogether ................ Thomas Beth 91
5-Cycle Systems with Holes ........................................... .
· ........................ Darryn E. Bryant, D. G. Hoffman, and C. A. Rodger 103
Stories about Groups and Sequences . . . . . . . . . . . . . . . . . . . . . .. Peter J. Cameron 109
Groups Admitting a Kantor Family and a Factorized Normal Subgroup ........ .
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Dirk Hachenberger 135
Spreads in Strongly Regular Graphs .................................... .
· .............................. Willem H. Haemers and Vladimir D. Tonchev 145
Codes Based on Complete Graphs ...................................... .
· ............ Dieter Jungnickel, Marialuisa J. de Resmini, and Scott A. Vanstone 159
A Construction of Partial Difference Sets in 7l,p2 x 7l,p2 X ... X 7l,p2 ....•.•••...
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Ka Bin Leung and Siu Lun Ma 167
On the Characterisation of AG(n, q) by its Parameters as a Nearly Triply Regular
Design. . . . . . . . . . Arlene A. Pascasio, Cheryl E. Praeger, and Blessilda P. Raposa 173
The Fundamental Theorem of q-Clan Geometry ................... S. E. Payne 181
Extension of Gravity Centers Configuration to Steiner Triple Systems .......... .
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Gunter Pickert 203
Constructions of Partial Difference Sets and Relative Difference Sets Using Galois
Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. D. K. Ray-Chaudhuri and Qing Xiang 215
m-Systems and Partial m-Systems of Polar Spaces ......................... .
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. E. E. Shult and J. A. Thas 229
Piotrowski's Infinite Series of Steiner Quadruple Systems Revisited ........... .
· ...................................................... Helmut Siemon 239
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Copyright © 1996 by Kluwer Academic Publishers
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Designs, Codes and Cryptography, 8, 7 (1996)
© 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Preface
The present special issue of Designs, Codes and Cryptography consists of some of the papers
which have been submitted (by invitation) in honour of Professor Dr. Hanfried Lenz who
celebrates his 80th birthday on April 22, 1996; one or two further such issues are to follow
later. Professor Lenz is well-known for his fundamental work in all parts of geometry; in
particular, his major interest in the last two decades has been in Finite Geometry and Design
Theory. For a detailed description of Prof. Lenz' research, I refer to the joint article by
Prof. G. Pickert and myself at the beginning of this issue. I take this opportunity to thank
all the authors of these special Lenz issues for their contributions and all the referees for
their (in most cases) speedy replies which made it possible to produce at least the first of
these issues well in time for Prof. Lenz' actual birthday and thus also for his 80th birthday
celebration to be held at the Freie Universitat Berlin on May 11, 1996. It only remains
to express my hope that Prof. Lenz will like the celebration as well as this special tribute
prepared for him and will continue to enjoy both good health and an ongoing interest in
mathematics.
Dieter lungnickel
Augsburg, September 7, 1995
Designs, Codes and Cryptography, 8, 9-22 (1996)
© 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
A Life's Work in Geometry:
An Homage to Hanfried Lenz
DIETER JUNGNICKEL
lnstitut for Mathematik, Universitiit Augsburg. D-86135 Augsburg. Germany
GUNTER PICKERT
Mathematisches lnstitut. lustus-Liebig-Universitiit Giessen. Amdtstr. 2, D-35392 Giessen. Germany
Dedicated to Hanfried Lenz on the occasion of his 80th birthday
Abstract. We give a brief review of the mathematical work of Hanfried Lenz, at the occasion of his 80th birthday
on Apri122, 1996.
Hanfried Lenz, who celebrates his 80th birthday on April 22, 1996, is without doubt one of
the leading geometers of this century. In what follows, we will give a review of his work
which consists of a list of his publications (about one hundred items) together with a brief
description of his results, where we will try to comment on nearly all of his papers. We
will organize this in a mixture between chronological and thematical arrangement. As the
reader will see, almost all of Prof. Lenz's research has been influenced by geometric points
of view; but here is a geometer with a range of interests which is of exceptional extent,
covering topics ranging from geometrical aspects of complex analysis via the foundations
of projective and affine geometry and the algebraic aspects (such as transformation groups
and quadratic forms) to convex and ordered geometry as well as finite geometries and design
theory and not even neglecting the pedagogical and didactic aspect of teaching geometry
(and mathematics in general).
The scientific publications of Hanfried Lenz started in 1950 with two shorter papers
concerning problems in analysis, more precisely certain trigonometrical sums and the com
putation of some integrals [1, 2]. Just four of his later papers belong likewise to (complex)
analysis; they concern the Schwarz polygonal mapping [3], Cramer asymptotic develop
ments [6], theta functions [20] and elliptic functions [25]. Already in 1952, we have the
transition from complex analysis to geometry in [4], where a surface consisting of a family
of circles in space is mapped conformally onto a sphere. The shift of emphasis may be seen
from Lenz's preface, where he states that his work avoids the use of the theory of conformal
invariants and uses (except for some elementary differential geometry and complex anal
ysis) as far as possible intuitive reasoning.' The same topic is considered also in another
paper [5] which appeared in 1952: Here families of spheres (in arbitrary dimensions) are
considered; the spheres are mapped conformally onto each other by orthogonal trajectories.
10 JUNGNICKEL AND PICKERT
In particular, the question of closed orthogonal trajectories is studied. These early analytic
publications grew out of Prof. Lenz's Ph.D. thesis of 1951.
In 1953, we find a first algebraic investigation [7] on finite automorphism groups of
infinite field extensions. Here the author's interest in projective geometry (one of the main
themes of his future research) is already evident when he proves the following geometric
consequence of his algebraic results: If the coordinatising field is a quadratically closed
field of characteristic 0 and if the 2n-th power of a collineation is projective (for some n),
=
then this in fact has to hold for n 1, too, i.e., already the square of the collineation is
projective. In the same year, we also find his first paper on projective planes [8] where he
exhibited the first example of a finite plane (in fact of order 16) containing both quadrangles
with collinear diagonal points and also quadrangles with non-collinear diagonal points, i.e.
both Fano and anti-Fano configurations.
In 1954, one of Prof. Lenz's most influential papers [13] appeared where he introduced the
classification of projective planes which was later named after him. Given a projective plane
IT, an incidence structure L(IT) is determined with the property that the little Desargues
theorem with centre P and axis g holds for a flag (P, g) of IT if and only if P and g are
incident in L(IT). Then there are only seven possibilities for L(IT), yielding the seven Lenz
classes of projective planes:
I. L(IT) is empty.
II. L (IT) consists of just one flag (P, g).
III. L (IT) contains the points of a line g and all lines through a point P ~ g.
IV. a) L(IT) consists of a line and all the points on it.
b) L(IT) consists of a point and all the lines on it.
V. L (IT) consists of the points of a line g and all lines through a point PEg.
VII. L(IT) equals IT.
In [13], two more classes (VI a, VI b) were listed which were subsequently proved to
be empty by San Soucie in 1955; for all the other classes, there are examples. The Lenz
classification proved to be a very fruitful concept, especially after it was refined by Barlotti
in 1957 who introduced the set LB(IT) of those pairs (P, g) in IT, for which the Desargues
theorem with centre P and axis g holds; thus the flags of IT in LB(IT) form the incidence
structure L(IT). Barlotti determined all possibilities for LB(IT), resulting in the celebrated
Lenz-Barlotti classification of projective planes which has become an indispensable tool
for the structural investigation of planes and which led to similar results for other types of
geometries, e.g. the Hering classification of Mobius planes.
Already in 1953, we see the start of Lenz's investigations on the foundation of affine
and projective geometries of arbitrary dimension d ~ 3 [9, 12, 14]. After constructing
the (not necessarily commutative) coordinate field, the traditional notation by coordinates
is used. As Lenz remarks, the objection that this means selecting an arbitrary coordinate
system can be answered by using a covariant notation with lower and upper indices for
point and hyperplane coordinates similar to the Ricci calculus.2 Some years later, in his
A LIFE'S WORK IN GEOMETRY: AN HOMAGE TO HANFRIED LENZ 11
didactically motivated paper [29] of 1958, Lenz discarded this point of view; by then, he felt
that the coordinate-free notation as preferred by Artin, Baer and Dieudonne is simpler and
more general than the traditional approach to linear algebra.3 A footnote concerning this
question gives a harsh criticism regarding his former papers in a way typical ofLenz and his
always self-critical attitude when he says that his older papers are an example of how not
to do it. 4 A particularly remarkable achievement of his paper [12] (which constituted Prof.
Lenz's "Habilitationsschrift") is an axiomatic treatment which uses hyperplanes (instead
of lines) as a fundamental concept; this results (under the restriction to finite dimensional
spaces) in a simple approach to the theory of polarity and to metric geometries. In [14], the
restriction to finite dimensions was discarded by introducing the notion of quasi-polarity; a
short supplement was given three years later in [26]. In [19], Prof. Lenz gave an interesting
synthetic definition of quadrics in a projective space of arbitrary dimension which is related
to the theory of polarity. In his paper [18], he considered the intersection of three quadrics
in 3-space; among other results, he proved that all points of intersection can be cubically
constructed provided that four of them are coplanar.
Prof. Lenz's papers [21, 22, 23, 24, 33] deal with topics in convex geometry. In [34]
which is characterized in the introduction as being mainly methodological he showed that
projective geometry provides a natural approach to some results on convex bodies and in
the affine theory of surfaces.s In connection with [24], the postulate of free mobility in the
real affine plane is analyzed in [27]. In [32], Lenz gave a characterization of the orthogonal
group in euclidean 3-space by axioms of free mobility which are simpler than those used
by Baer in 1950. This investigation was extended in [35] to cover linear semigroups in real
and complex n-space which have bounded eigenvalues.
The theory of polarity already mentioned before is a theme leading naturally to an analysis
of the concept of orthogonality [37, 43] and to the theory of quadratic forms which Lenz
studied in several papers [40, 44, 49, 50]. Following Witt, this theory is subsumed in the
theory of metric vector spaces. In [40], we find an application to finite geometries in the
form of a simplified treatment of the method of Dembowski (1958) and Hughes (1957)
for studying the collineation groups of finite projective planes. Of course, this method
generalizes the famous Bruck-Ryser method for studying the existence problem for finite
planes; a simplified proof of the Bruck-Ryser-Chowla theorem was to follow many years
later in [77].
The papers [36, 38, 55] deal with questions of ordering affine spaces by using the order
functions introduced by and named after Sperner. In [57], these order functions are again
used, this time for an introduction of the concept of angles which does not require the axiom
of parallels. Contributions to the foundation of the absolute geometry of space (connected
to Bachmann's reflection geometry) can be found in [41, 42, 59]. Reflection groups are
again considered in [62].
Many of Hanfried Lenz's results up to 1965 were incorporated in his book "Vorlesungen
tiber projektive Geometrie" [52]. In its preface, Lenz stated that projective geometry does
no longer offer many open problems, except for the still very active study of projective
planes.6 Nevertheless, he considered projective geometry indispensable for uniting many
classical geometric theories in the sense of Felix Klein's "Erlanger Programm", for the
foundation of non-Euclidean geometry and as a prior step for studying algebraic geometry.
12 JUNGNICKEL AND PICKERT
Lenz went on commenting on Dieudonn6's criticism of projective geometry (who consid
ered the area obsolete but still useful for the just mentioned purposes) and saying one should
not take this too seriously, since Dieudonne himself had contributed to projective geometry;
the classical methods and results of projective geometry should rather be included in the
modern treatment of mathematics.11t should be emphasized that Lenz's book does not only
provide a very nice way to introduce and develop the standard results (such as axiomatic
introduction of projective geometry, construction of coordinate fields, configuration the
orems, projective mappings, collineations and the like) but goes far beyond the scope of
introductory books. In particular, the chapters V (relations of separation and betweenness),
VII (quadratic forms over finite and p-adic fields and over the rational field), X (containing
an introduction ("Vorkurs") to algebraic geometry) and XI (a short introduction to topo
logical projective geometry, a topic to which Lenz made an important contribution in [51])
deserve explicit mentioning. These chapters indeed lead far beyond the classical results of
projective geometry (which are treated in Chapters I-IV and VI). Already one year later,
Lenz published a short monograph on non-Euclidean geometry [54]; this was intended as
an introductory text for students and accordingly published as an inexpensive (alas long
out-of-print) monograph in the well-known series of paperbacks by the Bibliographisches
Institut.
There is another, even earlier book by Hanfried Lenz, which is of a completely different
type, namely his "Grundlagen der Elementarmathematik" [39] which appeared in 1961.
It shares just one property with the two books already mentioned: it likewise stems out
of lecture courses given by Lenz at the Technical University of Munich. But this book
was explicitly addressed to teachers and future teachers ("Lehramtskandidaten") and is
intended to help this audience with the logical understanding of the questions related to
elementary mathematics. According to Lenz, elementary mathematics comprises on one
hand results which may be formulated and proved by using only particularly simple logical
tools and on the other hand those branches of mathematics which have been studied since
antiquity because of their close connection to natural phenomena. Accordingly, the book
contains three parts dealing with sets and numbers, elementary geometry, algebra and
coordinate geometry, respectively.s Altogether, the preface comprises seven pages and
demonstrates quite clearly that Hanfried Lenz belongs to the relatively small number of
research mathematicians who already in the early sixties were concerned about the training
and continuing education of teachers of mathematics. His book really fulfills the obligation
he felt about this task.
There are several further publications of Prof. Lenz which are aimed to the same goal
and would (in later terminology) be termed didactical ("stoffdidaktisch", to give the precise
German term). In his paper [29] on the foundations of analytic geometry which we have
already discussed before we find a sentence which is of fundamental importance both for
freshman courses at the university level and courses towards the end of high school (in
the German "Kollegstufe"), where Lenz remarks that linear algebra is the strongest tool
of analytic geometry but that it does not constitute analytic geometry by itselfY In [30],
Lenz gave an analysis of the Peano axiom system for the natural numbers, an important
piece of background knowledge which every teacher should have. For the same reason,
the axiomatic foundation of planar euclidean geometry based on the concept of reflections
