Integrable System and Motion of Curves in Projective and Similarity Geometries

Based on the natural frame in the projective geometry, motions of curves in projective geometry are studied. It is shown that several integrable equations including Sawada-Kotera and KK equations arise from motion of plane curves in projective geometries. Motion of space curves described by acceleratlon field and governed by endowing an extra space variable in similarity geometry P^3 is also studied.

CONSTRUCTION OF NONSYSTEMATIC LOW-DENSITY PARITY-CHECK CODES BASED ON SYMMETRIC BALANCED INCOMPLETE BLOCK DESIGN

This paper studies the nonsystematic Low-Density Parity-Check(LDPC)codes based onSymmetric Balanced Incomplete Block Design(SBIBD).First,it is concluded that the performancedegradation of nonsystematic linear block codes is bounded by the average row weight of generalizedinverses of their generator matrices and code rate.Then a class of nonsystematic LDPC codes con-structed based on SBIBD is presented.Their characteristics include:both generator matrices andparity-check matrices are sparse and cyclic,which are simple to encode and decode;and almost arbi-trary rate codes can be easily constructed,so they are rate-compatible codes.Because there aresparse generalized inverses of generator matrices,the performance of the proposed codes is only0.15dB away from that of the traditional systematic LDPC codes.

Two constructions of A^3-codes from projective geometry in finite fields

An A^3-code is extension of A^2-code in which none of the three participants： transmitter, receiver and arbiter, is assumed trusted. In this article, from projective geometry over finite fields, two A^3 -codes were given, the parameters, and probabilities of successful attacks were computed.

Single View Based Measurement on Space Planes

The plane metrology using a single uncalibrated image is studied in the paper, and three novel approaches are proposed. The first approach, namely key-line-based method, is an improvement over the widely used key-point-based method, which uses line correspondences directly to compute homography between the world plane and its image so as to increase the computational accuracy. The second and third approaches are both based on a pair of vanishing points from two orthogonal sets of parallel lines in the space plane together with two unparallel referential distances, but the two methods deal with the problem in different ways. One is from the algebraic viewpoint which first maps the image points to an affine space via a transformation constructed from the vanishing points, and then computes the metric distance according to the relationship between the affine space and the Euclidean space, while the other is from the geometrical viewpoint based on the invariance of cross ratios. The second and third methods avoid the selection of control points and are widely applicable. In addition, a brief description on how to retrieve other geometrical entities on the space plane, such as distance from a point to a line, angle formed by two lines, etc., is also presented in the paper. Extensive experiments on simulated data as well as on real images show that the first and the second approaches are of better precision and stronger robustness than the key-point-based one and the third one, since these two approaches are fundamentally based on line information.

Finite Projective Geometries and Classification of the Weight Hierarchies of Codes(Ⅰ)

The weight hierarchy of a binary linear [n, k] code C is the sequence (d 1, d 2, . . . , d k ), where d r is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries. The possible weight hierarchies in class A, B, C, D are determined in Part (I). The possible weight hierarchies in class E, F, G, H, I are determined in Part (II).

A Class of the Hamming Weight Hierarchy of Linear Codes with Dimension 5

The weight hierarchy of a [n, k; q] linear code C over Fq is the sequence （d1,…, dr,… , dk）, where dr is the smallest support weight of an r-dimensional subcode of C. In this paper, by using the finite projective geometry method, we research a class of weight hierarchy of linear codes with dimension 5. We first find some new pre- conditions of this class. Then we divide its weight hierarchies into six subclasses, and research one subclass to determine nearly all the weight hierarchies of this subclass of weight hierarchies of linear codes with dimension 5.