Showing papers in "Journal of Geometry in 1996"
TL;DR: The codeword estimating failure probability of a maximum likelihoodcodeword estimator for aq-nary symmetric channel does not depend on the transmitted codewORD, if the automorphism group of the code acts transitively on the set of codewords.
Abstract: LetF be a finite set of cardinality ¦F¦ =q ≥2,n ≥ 1 an integer and ϱ:F
n×Fn→ℕ0 theHamming metric. Acode isomorphism C → D between two block codesC,D
$$ \subseteq$$
Fn is defined as an isometry which can be extended to an isometry of the whole space Fn. Any permutation π ∈S
n
of the positions canonically induces a so-calledequivalence map
$$\tilde \pi$$
∈ Aut Fn; any system κ ≔ (κ1,κ2,...,κn) ofn permutations of the character setF induces a so-calledconfiguration
$$\tilde \kappa$$
∈ Aat Fn. The group Aut Fn of all isometries of Fn turns out to be semidirect product of the configuration group with the symmetric group of degreen. The codeword estimating failure probability of a maximum likelihood codeword estimator for aq-nary symmetric channel does not depend on the transmitted codeword, if the automorphism group of the code acts transitively on the set of codewords. When using a systematic (n, k)-encoder, the symbol decoding failure probability does not depend on the transmitted symbol or on the time of transmission if the configuration group and the automorphism group act transitively on the set of codewords resp. on the set of thek information positions.
25 citations
TL;DR: In this paper, the authors present two pencils of cubic curves that are the result of certain geometrical constructions in the triangle plane, and one of them turns out to be the probably most significant pencil of anallagmatic cubics that are associated with triangle geometry.
Abstract: The subject of this paper are two pencils of cubic curves that are the result of certain geometrical constructions in the triangle plane. One of them turns out to be the probably most significant pencil of anallagmatic cubics that are associated with triangle geometry. Both contain virtually all important single cubics, and other well known curves appear closely connected with them.
14 citations
TL;DR: In this paper, the authors give a complete and short proof of KAHN's Theorem that every locally projective space (M,M) with dim M ≥ 3 satisfying the Bundle Theorem is embeddable in projective spaces.
Abstract: We give a complete and short proof of KAHN's Theorem that every locally projective space (M,M) with dim M≥3 satisfying the Bundle Theorem is embeddable in a projective space. The central tool of KAHN's proof is the fact that (M,M) is locally projective, while we use mainly the Bundle Theorem.
14 citations
TL;DR: In this article, it was shown that there are only two complete 20-caps in PG(3,5) and determine their collineation groups up to isomorphism.
Abstract: Ak-cap in PG(3,q) is a set of k points, no three of which are collinear. A k-cap is calledcomplete if it is not contained in a (k+1)-cap. The maximum valuem2(3, q) ofk for which there exists a k-cap in PG(3,q) is q2+1. Letm2(3, q) denote the size of the second largest complete k-cap in PG(3,q). This number is only known for the smallest values of q, namely for q=2, 3,4 (cf. [2], pp. 96–97 and [3], p. 303). In this paper we show thatm2(3,5)=20. We also prove that there are, up to isomorphism, only two complete 20-caps in PG(3,5) and determine their collineation groups.
12 citations
TL;DR: In this article, a suitable "attraction" property is introduced to realize minimum blocking sets of a finite projective plane. But this property is not suitable for the case of infinite projective planes.
Abstract: Some results on configurations realizing minimum blocking sets of a finite projective plane are obtained by introducing a suitable “attraction” property.
10 citations
TL;DR: In this paper, it was shown that there exist complete k-caps in PG(3, q), q an odd prime ≥ 5 or q = 9, such that k = (q2 + q + 6)/3 or k = q2 + 2q+6)/3.
Abstract: The main problem on caps, posed originally by Segre in the fifties, is to determine the values of k for which there exists a complete k-cap. Very few results on this problem are known. The cardinality of the largest cap(s) and the smallest complete cap(s) are crucial. In this paper it is shown that there exist complete k-caps in PG(3, q), q an odd prime ≥ 5 or q = 9, such that k = (q2 + q + 6)/3 or k = (q2 + 2q + 6)/3. These complete caps are smaller than those currently known for q odd.
10 citations
TL;DR: In this article, it was shown that any derivation of a flock in PG(3,K), forK a full field, produces either a flock or a maximal partial flock.
Abstract: In this article, we construct some infinite analogues of the Fisher flocks. We also consider maximal partial flocks which may be constructed in several ways. In particular, we show that any derivation of a flock inPG(3,K), forK a full field, produces either a flock or a maximal partial flock. We provide constructions which produce maximal partial flocks or maximal partial spreads.
10 citations
TL;DR: In this article, the remarkable subsets of a semilinear space like cliques, anticliques, blocking sets and ovoids are studied, and they are shown to have a similar structure to ours.
Abstract: In this paper we study the remarkable subsets of a semilinear space like cliques, anticliques, blocking sets and ovoids.
9 citations
TL;DR: In this paper, at least three-dimensional partial affine spaces are characterized as partial linear spaces with parallelism fulfilling certain axioms, and the dimension of such spaces is defined to be equal to the degree of the corresponding affine space.
Abstract: An incidence structure with parallelism is said to be a partial affine space if it is embeddable in an affine space with the same pointset preserving the parallelism. Hence partial affine spaces are isomorphic to affine spaces, in which only complete parallel classes of lines are allowed to be missing. The dimension of a partial affine space is defined to be equal to the dimension of the corresponding affine space. In this article, at least three-dimensional partial affine spaces will be characterized as partial linear spaces with parallelism fulfilling certain axioms.
7 citations
TL;DR: In this paper, the complete list of k-arcsK in PG(n, q) fixed by a projective group isomorphic to A5 or A6, which acts primitively on the points of K, is presented.
Abstract: The complete list of thek-arcsK inPG(n, q) fixed by a projective group isomorphic toA5 orA6, which acts primitively on the points ofK, is presented. This leads to new classes of 10-arcs inPG(n, q), 3 ≤n ≤5. Our results also show that the non-classical 10-arc inPG(4, 9), discovered by D.G. Glynn [3], belongs to an infinite class of 10-arcs inPG(4, 3h),h ≥2, fixed by a projective group isomorphic toA6.
6 citations
TL;DR: As shown by MacWilliams the Hamming weight enumerators of a linear codeC over a finite field can be expressed by the weight enumerator of its dual codeC⊥ by using only elementary linear algebra.
Abstract: As shown by MacWilliams the Hamming weight enumerator of a linear codeC over a finite field can be expressed by the weight enumerator of its dual codeC⊥ A short inductive proof of this formula is given which uses only elementary linear algebra
TL;DR: In this article, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: "In any right triangle the hypotenuse is greater than the leg" and "If ∠AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is more than the segment BD".
Abstract: On the basis of the theorye− of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: “In any right triangle the hypotenuse is greater than the leg” and “If ∠AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD.” This represents an attempt to split the Pasch axiom with respect toe−. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.
TL;DR: In this article, it was shown that a partial flock of a quadratic cone in PG(3,q) which properly contains a linear subflock of at least (q−1)/2 conics may be uniquely extended to a linear or Fisher flock.
Abstract: It is shown that a partial flock of a quadratic cone in PG(3,q) which properly contains a linear subflock of at least (q−1)/2 conics may be uniquely extended to a linear or Fisher flock.
TL;DR: In this article, the authors determine all 4-dimensional compact projective planes with a solvable 6-dimensional collineation group fixing two distinct points, and acting transitively on the affine pencils through the fixed points.
Abstract: We determine all 4-dimensional compact projective planes with a solvable 6-dimensional collineation group fixing two distinct points, and acting transitively on the affine pencils through the fixed points. These planes form a 2-parameter family, and one exceptional member of this family is the dual of the exceptional translation plane with 8-dimensional collineation group.
TL;DR: Using suitable subgroups of Singer cyclic groups, this article proved some properties of regular spreads and Segre varieties, which in turn yield a necessary and sufficient condition for partitioning a finite projective space into such varieties.
Abstract: Using suitable subgroups of Singer cyclic groups we prove some properties of regular spreads and Segre varieties, which in turn yield a necessary and sufficient condition for partitioning a finite projective space into such varieties.
TL;DR: In this paper, the linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point.
Abstract: It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of real inner product spaces among complete, convex, externally convex metric spaces is obtained.
TL;DR: Lu and Jinh-Zhong as mentioned in this paper generalized the Neuberg-Pedoe inequality to R n by applying Maclaurin's inequality to a determinant equation.
Abstract: Yang Lu and Zhang Jinh-Zhong generalized the Neuberg-Pedoe inequality toR n by applying Maclaurin's inequality to a determinant equation. We noticed that a simple proof of an equivalent formula follows from their positive definite matrices alone. The resulting formula also has a simple matrix formulation.
TL;DR: In this paper, a subgroup of the general linear group GLn(K) is generated by the setS of all elements σ inG for which dimV(σ − 1) = 1.
Abstract: LetG be a subgroup of the general linear group GLn(K), where charK ≠ 2. Put Kn =V. AssumeG is generated by the setS of all elements σ inG for which dimV(σ − 1) = 1, and suppose σ2=1V for each σ inS. If {V(σ−1)¦σ∈S} contains a simplex, if − 1V ∈G, and if π inG is a product of dim v(π−1) elements σ inS wheneverV(π−1) is not contained in the kernel ofπ−1, thenG is a subgroup of an orthogonal group.
TL;DR: In this article, the existence of shape-regular polygons and the correspondence between shape regularity and affine regularity were investigated in the affine plane, where the Gaussian plane was generalized to affine planes.
Abstract: The notion of shape in the Gaussian plane was introduced by Lester [5] and extended by Artzy [1]. In this paper we generalize this notion in the affine planesAG(2,q) over the Galois fieldGF(q), q=p
r andp an odd prime. We investigate the existence of shape-regular polygons and the correspondence between shape-regularity and affine-regularity.
TL;DR: In this article, a class of finite planar spaces whose planes have two sizes is characterized, and the authors show that these spaces can be characterized in terms of two-dimensional spaces.
Abstract: In this paper we characterize a class of finite planar spaces whose planes have two sizes.
TL;DR: In this paper, the volume of simplices with edges of equal length in a Banach space is studied and natural generalizations of planar results on volume ratios with respect to convex bodies in finite dimensional Banach spaces are given.
Abstract: Natural generalizations of planar results on volume ratios with respect to convex bodies in finite dimensional Banach spaces are given. The underlying paper contains inequalities for the volume of simplices with edges of equal length in a Banach space.
TL;DR: In this article, a combinatorial characterization of the Klein quadric in terms of its incidence structure of points and lines is given, and a proof of a result of Havlicek is given.
Abstract: We give a combinatorial characterization of the Klein quadric in terms of its incidence structure of points and lines. As an application, we obtain a combinatorial proof of a result of Havlicek.
TL;DR: In this article, a complete classification of finite matroids with one more hyperplane than points is given, and a classification of matroid classes with more than one hyperplane is presented.
Abstract: In this paper a complete classification of finite matroids with one more hyperplane than points is obtained.
TL;DR: In this article, the authors embed a family of free constructions, called the freelocally affine planes, into the discipline of partial order with noticeable technical simplifications as a consequence, which is a follow up of (3) which introduces free geometries in terms of simple extensions.
Abstract: The number of types of free constructions introduced into geometry has increased steadily since the birth (2) of the free projective planes in 1943. The present paper is a follow up of (3) which introduces free geometries in terms ofsimple extensions. This point of view embeds a family of such constructions, here called the freelocally affine planes, into the discipline ofpartial order with noticeable technical simplifications as a consequence.
TL;DR: In this paper, the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature was investigated.
Abstract: We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature Also, if∼M is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection of∼M is quasi-Einstein, too, provided thatM is tangent to the Lee field of∼M As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ
m
(of a complex Hopf manifoldS
2m+1 ×S
1)
TL;DR: The affine chain geometry over a group with a partial fibration into subgroups and a certain involution is introduced in this paper, which generalizes the affine trace of the chain geometries over an associative algebra.
Abstract: The affine chain geometry over a group with a partial fibration into subgroups and a certain involution is introduced. This concept generalizes the affine trace of the chain geometry over an associative algebra. We study the geometric properties of these geometries and give examples.
TL;DR: In this article, the authors defined finite mean type hypersurfaces to be hypersurface with mean curvature function of finite Chen-type and proved that hyperplanes and cylinders on spheres are the only polynomial translation hypersurveys of finite mean-type in a Euclidean space.
Abstract: We define finite mean type hypersurfaces to be hypersurfaces with mean curvature function of finite Chen-type. Then, we prove that hyperplanes are the only polynomial translation hypersurfaces of finite mean type in a Euclidean spaceEn+1. And we show that the only non-conic hyperquadrics of finite mean type in Euclidean spaces are the hyperspheres and the cylinders on spheres. Finally, we state that, among all hypercylinders in a Euclidean spaceEn+1, the only ones of finite mean type are those on finite mean type planar curves.
TL;DR: In this article, it was shown that the conjunction of the axioms (R1 and R2) is weaker than (R), i.e., if we replace (R) by (R 1) and (R 2) in their axiom system, we get an axiomatic system for a larger class of partial affine spaces.
Abstract: S.Meuren and A.Herzer have introduced in [1] an axiom (R). We show that the conjunction of the axioms (R1) and (R2) is weaker than (R), i. e., if we replace (R) by (R1) and (R2) in their axiom system, we get an axiom system for a larger class of partial affine spaces.