THREE-DIMENSIONAL LINEAR CODES AND COORDINATED FINITE PROJECTIVE PLANE

SADIKI, Flamure and IBRAIMI, Alit and SALIHI, Ylldrita and XHAFERI, Miranda (2022) THREE-DIMENSIONAL LINEAR CODES AND COORDINATED FINITE PROJECTIVE PLANE. Journal of Natural Sciences and Mathematics of UT, 7 (13-14). pp. 121-129. ISSN 2671-3039

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Abstract

In this paper, we study the connections between linear codes and projective geometries over finite fields. Each of these two topics is interesting by itself and has been subject to substantial research. In the last decade, a lot of progress has been made in both areas. We introduce some of the basic ideas and connections between finite projective spaces and coding theory. We begin by studying projective geometries, from this, we introduce a very interesting action in projective planes which lead to many other interesting areas of finite geometry, coordination of the plane. We will coordinate the lines using point coordination. Our focus then shifts to coding theory and in particular three-dimensional linear codes. The linear code C_(s,t) n,q of s-spaces and t-spaces in a projective space PG (n,q),q=p^d, p prime, is defined as the vector space spanned over Fqby the rows of the incidence matrix of s-spaces and t-spaces. Three-dimensional code applied on the constructed projective model: Fano Plane, like a3-dimensional vector space over F2. The Fano plane, like a model, occurs in algebraic geometry and geometric algebra in a number of cases, constructing a link between such important mathematical concepts. There are given different ways of constructing the model, taking in consider that it is impossible to label the Fano plane in such a way that all or just five of its lines would be ordinary.

Item Type:	Article
Uncontrolled Keywords:	Finite Projective space, Fano plane, Coordination, Vector space, Generator matrix, Linear codes
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Engineering, Science and Mathematics > School of Mathematics
Depositing User:	Unnamed user with email zshi@unite.edu.mk
Date Deposited:	31 Oct 2022 13:03
Last Modified:	31 Oct 2022 13:03
URI:	http://eprints.unite.edu.mk/id/eprint/1067

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