Finite projective geometries and linear codes
WebFeb 1, 2000 · Linear codes over finite fields and finite projective geometries February 2000 Authors: Ivan N. Landjev Abstract The aim of this paper is to survey relationships … WebJul 1, 2024 · The linear representation of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a direct geometric interpretation in terms of certain hyperplanes.
Finite projective geometries and linear codes
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WebJul 1, 2024 · Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are … WebThis paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite …
WebLinear Algebra and Geometry - Dec 09 2024 This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner WebMay 1, 2024 · The explanation for this behavior comes from the finite geometries. In the geometry over Z / 〈 6 〉, the points ( 0, 0) and ( 3, 0) are neighbors contained in three distinct lines. We get d ( C S 0 ( F 7)) ≤ 36 − 3 ⋅ 6 = 18 from Corollary 3.3. In the plane over Z / 〈 8 〉, on the other hand, the situation is more subtle.
WebOct 16, 2024 · The Geometric Approach to Linear Codes, Finite Geometries (Proc. 4th Isle of Thorns Conf, Chelwood Gate, UK, July 16–21, 2000), Blokhuis A., Hirschfeld, J.W.P., … WebApr 5, 2012 · In this article, several new constructions for ring-linear codes are given. The class of base rings are the Galois rings of characteristic 4, which include $${\\mathbb {Z}_4}$$ as its smallest and most important member. Associated with these rings are the Hjelmslev geometries, and the central tool for the construction is geometric dualization. …
WebAug 19, 2024 · Title:Linear representations of finite geometries and associated LDPC codes Authors:Peter Sin, Julien Sorci, Qing Xiang Download PDF Abstract:The {\it …
WebNov 20, 2024 · An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear … memes of the boysWebIn finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite … memes of the northWebLinear Codes over Finite Chain Rings and Projective Hjelmslev Geometries. Linear codes over finite chain rings correspond to multisets of points in finite projective Hjelmslev … memes of thanksgivingWebThe first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. memes of thelma and louiseWeb(LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Eu-clidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth 6. Finite geometry LDPC codes can be decoded in var- memes of thank youWebApr 7, 2009 · The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. memes of the decadeWebLINEAR REPRESENTATIONS OF FINITE GEOMETRIES AND ASSOCIATED LDPC CODES. PETER SIN, JULIEN SORCI AND QING XIANG Abstract. The linear representation of a subset of a nite projective space is an incidence system of a ne points and lines determined by the subset. In this paper we use character theory to show that … memes of the month