130,485 research outputs found
Affine attenuated spaces
No full textWe prove that a rank d geometry satisfying the “Intersection Property” and belonging to the diagram with s > 3 and d ⩾ 4 is isomorphic to an affine attenuated space (i.e., the geometry of the subspaces of a projective space not meeting a given subspace and not contained in a fixed hyperplane)
Remarks on polarity designs
No full textJungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140) used polarities
of PG(2d − 1, q) to construct non-classical designs with a hyperplane and the same
parameters and same intersection numbers as the classical designs PGd (2d, q), for every
prime power q and every integer d ≥ 2. Our main result shows that these properties already
characterize their polarity designs. Recently, Jungnickel and Tonchev introduced new invariants for simple incidence structures D, which admit both a coding
theoretic and a geometric description. Geometrically, one considers embeddings of D into
projective geometries = PG(n, q), where an embedding means identifying the points of
D with a point set V inin such a way that every block of D is induced as the intersection of
V with a suitable subspace of . Then the new invariant—which we shall call the geometric
dimension geomdimq
D of D—is the smallest value of n for which D may be embedded into
the n-dimensional projective geometry PG(n, q). The classical designs PGd (n, q) always
have the smallest possible geometric dimension among all designs with the same parameters,
namely n, and are actually characterized by this property. We give general bounds for
geomdimq
D whenever D is one of the (exponentially many) “distorted” designs constructed
in Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140]
Some Geometric Aspects of Finite Abelian Groups
No full textIn this survey we show how, abelian collineation groups of projective planes, may be applied for the construction of interesting geometric objects such as unitals, arcs and (hyper-)ovals, (Baer) subplanes, and projective triangles. This approach makes it sometimes possible to provide simple geometric proofs for non-trivial structural restrictions on the given collineation group
Panmagic Sudoku
No full textWe show how to embed any given magic square of order n into a Sudoku square of order n ^2 for which the n diagonal blocks consist of the given magic square. Moreover, our construction proves that the existence of a panmagic square of order n implies that of a Sudoku square of order n^2 for which all n^2 regions are panmagic
Projective planes with a large quasi-regular collineation group
No full textLondon Mathematical Society Lecture Note Series n. 307
Let be a finite projective plane of order n, and let G be a large
abelian (or, more generally, quasiregular) collineation group of ; to
be specific, we assume |G| > (n2 + n + 1)/2. Such planes have been
classified into eight cases by Dembowski and Piper in 1967. We survey
the present state of knowledge about the existence and structure of such
planes. We also discuss some geometric applications, in particular to the
construction of arcs and ovals. Technically, a recurrent theme will be
the amazing strength of the approach using various types of difference
sets and the machinery of integral group rings
On disjoint (3t,3,1) difference families
No full textWe give a new and easier proof of the existence of a disjoint (3t, 3, 1) cyclic difference family for every t >3, ?rst proved by Dinitz and Shalaby (2002). Our purely theoretical construction is still elementary but simpler and does not need to be checked by computer
PREFACE to the Special Issue Dedicated to Dan Hughes for his 80th birthday, (guest editors D.Ghinelli, J.Hirschfeld and D. Jungnickel)
No full text(the volume edited includes 28 research papers for a total of 309 pages
CODES FROM INCIDENCE MATRICES AND LINE GRAPHS OF PALEY GRAPHS
No full textWe examine the p-ary codes from incidence matrices of Paley
graphs P (q) where q ≡ 1 (mod 4) is a prime power, and show that the codes are [ q(q−1) /4 , q − 1, (q −1)/ 2 ]_2 or [ q(q −1) /4 , q, (q −1) /2 ]p for p odd. By finding PD-sets we
show that for q > 9 the p-ary codes, for any p, can be used for permutation decoding for full error-correction. The binary code from the line graph of P (q) is shown to be the same as the binary code from an incidence matrix for P (q)
Piani proiettivi finiti e neo-insiemi di differenze
No full texthttp://www.mat.uniroma1.it/~combinat/quadern
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