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Finite Projective Geometries and Linear Codes

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Finite Projective Geometries and Linear Codes Finite Projective Geometries and Linear Codes Tom Edgar Advisor: Dr. Anton Betten Department of Mathematics In partial fulfillment of the requirements for the degree of Master Science Fort Collins, Colorado Spring 2004 Abstract In this paper, we study the connections between linear codes and projective geometries over finite fields. Often good codes come from interesting structures in projective geometries. For example, MDS codes come from arcs (i.e. sets of points which are extremal in the sense that they admit no other than the obvious dependencies). In addition, we take a closer look at ovals and hyperovals in projective planes and ovoids in projective 3-spaces. In particular, we examine Glynn’s condition for the existence of hyperovals. Contents 1 Introduction 3 2 Projective Geometry 3 2.1 A Brief Introduction and Basic Definitions . 3 2.1.1 Quadratic Forms: Classification . 6 2.2 Arcs,Caps,andNormalRationalCurves . 8 2.3 Ovals and Hyperovals in PG2(q)................. 9 2.3.1 Glynn’s Condition for Existence of Hyperovals . 14 2.4 k-arcs, k-caps and minihypers in PGn(q)............ 25 2.4.1 k-arcsingeneral ..................... 25 2.4.2 A Glimpse into the World of k-caps: Ovoids . 29 2.4.3 Minihypers ........................ 34 3 Coding Theory 35 3.1 BasicDefinitions ......................... 35 3.2 MDSCodes ............................ 39 4 From Geometry to Linear Codes 42 4.1 Results: ArcstoMDSCodes. 42 4.1.1 Minihypers and the Griesmer Bound . 44 4.2 Conjectures ............................ 45 5 Conclusion 46 2 1 Introduction When coding theory was first introduced, the connections to projective ge- ometry were not known and hence not studied. Only recently have there been important advances in the connections between projective geometry and coding theory. In this paper, we introduce some of the basic ideas and connections between finite projective spaces and coding theory. We begin by studying projective geometries in general and extend this to studying certain incidence structures called arcs inside of projective geometries. From this, we introduce some very interesting structures in projective planes which lead to many other interesting areas of finite geometry. Our focus then shifts to cod- ing theory and in particular MDS codes. The main desire in coding theory is to get a great deal of information across a noisy communication channel with only short codewords and a large distance between them. MDS codes turn out to be one possible type of optimal codes and hence are of interest. We attempt to describe some of the results in this area. In addition, we examine the connection between MDS codes and arcs in projective geometries. We intend to introduce at least some of the main open problems and conjectures in this area. We assume a basic graduate level of math is known, and hope to include most of the relevant information in this area. The paper is a compi- lation of existing results, and we attempt to bring these together. We intend for the paper to read like a textbook so that it may be used as a resource for someone interested in researching and studying projective geometry and coding theory. 2 Projective Geometry 2.1 A Brief Introduction and Basic Definitions To begin we must first introduce some basic terminology with respect to d finite geometry. We consider only finite fields, Fq, where q = p , p prime. We assume some knowledge of the structure of finite fields and vector spaces F× F is known. Let q be the set of non-zero elements of q. Definition 2.1.1 A finite geometry is a pair G = (Ω, I) where Ω is a finite set and I is a relation on Ω that is symmetric and reflexive. I is called an incidence relation on Ω. 3 Definition 2.1.2 A projective space of dimension n over a field Fq is Fn+1 the set of non-zero subspaces of q with respect to inclusion. We denote this PGn(Fq), also called PGn(q). Remark 2.1.1 PGn(q) is a finite geometry with Ω being the set of non-zero Fn+1 subpaces of q and I is symmetric inclusion. We call the one dimensional Fn+1 Fn+1 subspaces of q the points, the two dimensional subspaces of q the Fn+1 lines, and the n-dimensional subspaces of q the hyperplanes of the ge- ometry. There are always 1+ q + ... + qn points and each line contains q +1 points. F3 Example 2.1.1 PG2(2) is the non-zero subspaces of 2. This is the projec- tive plane of order 2. There are 1+2+22 = 7 points, each a vector in the vector space. There are 3 points on every line. This structure is known as the Fano Plane and is described pictorially in a nice way in Figure 1: (1,0,0) [0,1,1] [0,0,1] (1,0,1) [1,1,1] (1,1,0) [0,1,0] (1,1,1) [1,0,1] (0,1,0) (0,0,1) (0,1,1) [1,0,0] [1,1,0] Figure 1: The Fano Plane PG2(2). This is the 3 dimensional vector space over F2. We label the points with parentheses and the lines with square brackets. We point out that even though lines look like they cross, they do not actually intersect unless they meet at a point. 4 Definition 2.1.3 The points a1, a2, ..., am in PGn(q) are collinear if there exists a line that contains every point ai. We say a point p = (x0, ..., xn) is on a line L = [y0, ..., yn] if and only if x0y0 + x1y1 + ... + xnyn = 0. There is a more general definition of a projective geometry. It turns out that all projective spaces with n ≥ 3 are of the form PGn(q) for q a prime power. However, in the special case of n = 2, i.e. the projective planes, there are Fn+1 certain examples that do not arise from the structure of q . In this case, we do include the general definition of a projective plane. Definition 2.1.4 A projective plane of order n is a set (P,B,I), where P is the set of points, B the set of lines, and I the incidence relation between them. The number of points is n2 +n+1 and the number of lines is n2 +n+1. Any line has n +1 points on it and there are n +1 lines on any point. In addition, we require that any two points determine a unique line and that any two lines intersect in a unique point. Projective planes in general form an interesting area of study. One of the main conjectures in the area is that projective planes of order n exist if and only if n is a prime power. It has been shown that projective planes of order 6 and order 10 do not exist. The case n = 12 is still open. Though there are many fascinating aspects to these general projective planes, from now on, we will only be considering projective spaces of the form PGn(q) as defined above. Like many disciplines in mathematics, we can learn a great deal about a structures in projective space by studying the automorphisms of a projective geometry. Definition 2.1.5 A collineation of a projective geometry is a permutation of the points such that the lines are mapped onto lines, and thus subpaces are mapped to subspaces. Definition 2.1.6 We consider the set of all invertible semilinear transfor- mations of PGn(q). This set forms a group under composition called ΓL(n + 1, q). Now, if H is the pointwise stabiliser of PGn(q), then H is a subgroup of ΓL(n +1, q) and we get that P ΓL(n +1, q)=ΓL(n +1, q)/H is called the projective semilinear group. 5 The set of collineations of a projective geometry forms a group called the automorphism group, denoted by AutPGn(q). Theorem 2.1.1 (cf. [1]) Let n ≥ 2. Then AutPGn(q)= P ΓL(n +1, q). This theorem is known as the Fundamental Theorem of Projective Ge- ometry. Baer has a nice proof of this in [1]. A good source of information on the collineations of a projective geometry is [15]. The idea of automorphisms of the projective geometry will allow us to decide if structures are unique up to isomorphism. Definition 2.1.7 Two sets of points in PGn(q) are called projectively equivalent if there exists an element α ∈ P ΓL(n +1, q) such that α maps one set to the other. 2.1.1 Quadratic Forms: Classification Before we can introduce and discuss quadratic forms, we first introduce bi- linear forms. Fn Fn Fn F Definition 2.1.8 A bilinear form B on q is a map from q × q → q defined by (x, y) → B(x, y) such that: 1. B(x + x′,y)= B(x, y)+ B(x′,y) 2. B(x, y + y′)= B(x, y)+ B(x, y′) 3. B(ax, y)= aB(x, y)= B(x, ay) For our purposes, we will only consider symmetric bilinear forms. These are the forms which have the special property that B(x, y) = B(y, x) for all x, y. We restrict ourselves to this case since when looking at the second definition of a quadratic form, we find that the bilinear form involved will be a symmetric one. So, from this point on, we only consider that B is symmetric. Definition 2.1.9 A quadratic form is a polynomial function arising from a homogeneous polynomial of degree 2 in n-variables.
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