The relative generalized Hamming weight of linear q-ary codes and their subcodes

In this article, some properties of the relative generalized Hamming weight (RGHW) of linear codes and their subcodes are developed with techniques in finite projective geometry. The relative generalized Hamming weights of almost all 4-dimensional q-ary linear codes and their subcodes are determined.      

一类代数几何码的构造

利用有限域Fq^8(s≥1为正整数，q为素数幂)上代数曲线构造了一类q元线性码，这类线性码是q^8元几何Goppa码的子域子码的子码，同时也是Chaoping Xing，SanLing构造的代数几何码[1]的推广。     

Subcodes of the Projective Generalized Reed-Muller Codes Spanned by Minimum-Weight Vectors

We use methods of Mortimer [19] to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals. These methods provide a proof, alternative to a dimension argument, that neither the projective generalized Reed-Muller code of order r and of length over the finite field F _q of prime-power order q, nor its dual, is spanned by its minimum-weight vectors for 0<r<m–1 unless q is prime. The methods of proof are the projective analogue of those developed in [17], and show that the codes spanned by the minimum-weight vectors are spanned over F _q by monomial functions in the m variables. We examine the same question for the subfield subcodes and their duals, and make a conjecture for the generators of the dual of the binary subfield subcode when the order r of the code is 1.     

Weight enumeration of codes from finite spaces

We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a finite projective or affine space. As a result from the geometric method we use for the weight enumeration, we also completely determine the set of supports of subcodes and words in an extension code.     

Projective deformations of hyperbolic Coxeter 3-orbifolds

By using Klein??s model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev??s theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahedra.     

Geometrical contributions to secret sharing theory

Finite geometry has found applications in many different fields and practical environments. We consider one such application, to the theory of secret sharing, where finite projective geometry has proved to be very useful, both as a modelling tool and as a means to establish interesting results. A secret sharing scheme is a means by which some secret data can be shared among a group of entities in such a way that only certain subsets of the entities can jointly compute the secret. Secret sharing schemes are useful for information security protocols, where they can be used to jointly protect cryptographic keys or provide a means of access control. We review the contribution of finite projective geometry to secret sharing theory, highlighting results and techniques where its use has been of particular significance.     

Strongly prime near-rings 2

The spliting systems of a finite group are used to induce a geometry associated with the group. The method generalizes the classical approach used to induce a geometry associated with a finite dimensional vector space and extends concepts related to the special and projective linear groups to arbitrary finite groups. Applications are made to finite solvable nC-groups and to the automorphism group of homocyclic abelian p-groups.     

Codes projectifs à deux poids, “caps” complets et ensembles de différences

Using certain sets of points of a finite projective geometry some results are obtained from properties of two-weight projective codes. A problem concerning complete caps is solved. A deep connection between binary uniformily packed codes and difference sets over elementary Abelian 2-groups is established and a characterization of these difference sets is given.     

The Permutation Modules for GL(n+1, Fq) Acting on Pn(Fq) and Formula

The paper studies the permutation representations of a finitegeneral linear group, first on finite projective space and thenon the set of vectors of its standard module. In both casesthe submodule lattices of the permutation modules are determined.In the case of projective space, the result leads to the solutionof certain incidence problems in finite projective geometry,generalizing the rank formula of Hamada. In the other case,the results yield as a corollary the submodule structure ofcertain symmetric powers of the standard module for the finitegeneral linear group, from which one obtains the submodule structureof all symmetric powers of the standard module of the ambientalgebraic group.     

基于新精确函数的区间直觉模糊多属性决策方法

基于区间直觉模糊数隶属度和非隶属度构成的二维几何图形特征给出区间直觉模糊数精确函数的新定义,并将其作为区间直觉模糊数的排序指标,区间直觉模糊数的精确函数值越大,则区间直觉模糊数就越大,进而提出一种权重信息不完全确定的区间直觉模糊多属性决策方法.通过算例分析说明所提出排序指标的有效性和决策方法的可行性.     

Inbreeding depression in a zygotic algebra

We study the algebraic behavior of a three dimensional zygotic algebra in the presence of parameters 0 < s < 1 and 0 < g < 1; s for selfing and g which reflects its associated inbreeding depression. We also study the dynamics of the system for which this algebra is a model. Our methods lean towards commutative algebra and algebraic geometry and find support on the computer program Macaulay2.Our results are best understood through the geometry of a rational function of the projective plane.     

A remarkable configuration: A generalization of computergenerated results by S.-C. Chou and its connection with a plane figure by A. Emch

The paper's starting point are four theorems on conics which can be found in a collection of computer proved results by C.-S. Chou from 1987. It not only contains a generalization of two of Chou's results but also a plane figure consisting of points, lines and conics. A suitable notation will reveal a striking symmetry of this figure. Moreover, it turns out that a plane figure from 1940 found by A. Emch using algebraic methods is very similar to ours, which we obtained synthetically. As an application in finite geometry we have gone some way towards regarding our figure as a real projective model of the finite projective plane of order 4.Dedicated to Dr. J. F. Rigby on the occasion of his 65th birthday     

The concept of physical and fractal dimension II. The differential calculus in dimensional spaces

The projective dimensional analysis based on the projective extension of scaling group and projective dimensional function is studied. The differential calculus corresponding to geometry of dimensional spaces is constructed and examined. At the next step we explore the projective extension of dimensional derivatives. Simple fractal models of various processes with changing fractal dimension illustrate the proposed methods.     

A unified approach to projective lattice geometries

The interest in pursuing projective geometry on modules has led to several lattice theoretic generalizations of the classical synthetic concept of projective geometry on vector spaces.Introduced in this paper is an approach that is capable of unifying various attempts within a new conceptual frame. This approach reflects algebraic properties from a lattice-geometric point of view. Together with new results we are presenting results from previous publications which have been improved in the frame of this work.     

A class of pbib-designs obtained from projective geometries

In this paper, a new class of partially balanced incomplete block designs is constructed over an association scheme obtained from a finite projective geometry. Further, a general method for deriving a balanced incomplete block design from another one is given.     

Extension theorems for linear codes over finite fields

We survey recent results on the extendability of linear codes over finite fields with link to projective geometry and some applications to optimal linear codes problem.     

Minimum Weight and Dimension Formulas for Some Geometric Codes

The geometric codes are the duals of the codes defined by the designs associated with finite geometries. The latter are generalized Reed–Muller codes, but the geometric codes are, in general, not. We obtain values for the minimum weight of these codes in the binary case, using geometric constructions in the associated geometries, and the BCH bound from coding theory. Using Hamada's formula, we also show that the dimension of the dual of the code of a projective geometry design is a polynomial function in the dimension of the geometry.     

A memoir on the projective geometry of spinors

Summary The classical methods of projective geometry are applied to a number of questions in general relativity, by using the Van der Waerden spinor analysis. These include a new geometric theory of spinors, refinements in the spinor calculus, the classification of electromagnetic and gravitational fields, Weyl-Maxwell fields, a classification of the Einstein spinor, and the projective geometry of the Bel-Petrov types. Dedicated to the memory of my teacher and friend Professor Dr.Vaclav Hlaváty (1894–1969) Entrata in Redazione il 14 marzo 1975.     

A conjecture of Beauville and Catanese revisited

A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic.     

Some remarks on orders of projective planes,planar difference sets and multipliers

In this paper we investigate how finite group theory, number theory, together with the geometry of substructures can be used in the study of finite projective planes. Some remarks concerning the function v(x)= x ² + x + 1are presented, for example, how the geometry of a subplane affects the factorization of v(x). The rest of this paper studies abelian planar difference sets by multipliers.Partially supported by NSA grant MDA904-90-H-1013.