Projective Geometries over Finite Fields

Authors

J. W. P. Hirschfeld

Type: Book

Publication Date: 1998-01-08

Citations: 1535

DOI: https://doi.org/10.1093/oso/9780198502951.001.0001

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Abstract

Abstract This book is an account of the combinatorics of projective spaces over a finite field, with special emphasis on one and two dimensions. With its successor volumes, Finite projective spaces over three dimensions (1985), which is devoted to three dimensions, and General Galois geometries (1991), on a general dimension, it provides a comprehensive treatise of this area of mathematics. The area is interesting in itself, but is important for its applications to coding theory and statistics, and its use of group theory, algebraic geometry, and number theory. This edition is a complete reworking of the first edition. The chapters bear almost the same titles as the first edition, but every chapter has been changed. The most significant changes are to Chapters 2, 10, 12, 13, which respectively describe generalities, the geometry of arcs in ovals, the geometry of arcs of higher degree, and blocking sets. The book is divided into three parts. The first part comprises two chapters, the first of which is a survey of finite fields; the second outlines the fundamental properties of projective spaces and their automorphisms, as well as properties of algebraic varieties and curves, in particular, that are used in the rest of the book and the accompanying two volumes. Parts II and III are entirely self-contained; all proofs of results are given. The second part comprises Chapters 3 to 5. They cover, in an arbitrary dimension, the properties of subspaces such as their number and characterization, of partitions into both subspaces and subgeometries, and of quadrics and Hermitian varieties, as well as polarities. Part III is a detailed account of the line and the plane. In the plane, fundamental properties are first revisited without much resort to the generalities of Parts I and II. Then, the structure of arcs and their relation to curves is described; this includes arcs both of degree two and higher degrees. There are further chapters on blocking sets and on small planes, which means of orders up to thirteen. A comprehensive bibliography of more than 3000 items is provided. At the end of each chapter is a section, Notes and References, which attributes proofs, includes further comments, and lists every relevant reference from the bibliography.

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Finite Geometry and Combinatorial Applications

2015-04-30

Simeon Ball

The projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs, codes … The projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs, codes and graphs. This book provides an introduction to these geometries and their many applications to other areas of combinatorics. Coverage includes a detailed treatment of the forbidden subgraph problem from a geometrical point of view, and a chapter on maximum distance separable codes, which includes a proof that such codes over prime fields are short. The author also provides more than 100 exercises (complete with detailed solutions), which show the diversity of applications of finite fields and their geometries. Finite Geometry and Combinatorial Applications is ideal for anyone, from a third-year undergraduate to a researcher, who wishes to familiarise themselves with and gain an appreciation of finite geometry.

Finite Geometries

2019-07-26

György Kiss , Tamás Szőnyi

Finite Geometries stands out from recent textbooks about the subject of finite geometries by having a broader scope. The authors thoroughly explain how the subject of finite geometries is a … Finite Geometries stands out from recent textbooks about the subject of finite geometries by having a broader scope. The authors thoroughly explain how the subject of finite geometries is a central part of discrete mathematics. The text is suitable for undergraduate and graduate courses. Additionally, it can be used as reference material on recent works. The authors examine how finite geometries' applicable nature led to solutions of open problems in different fields, such as design theory, cryptography and extremal combinatorics. Other areas covered include proof techniques using polynomials in case of Desarguesian planes, and applications in extremal combinatorics, plus, recent material and developments. Features: Includes exercise sets for possible use in a graduate course Discusses applications to graph theory and extremal combinatorics Covers coding theory and cryptography Translated and revised text from the Hungarian published version

Finite Geometries and Combinatorics

1993-10-21

Frank De Clerck , Albrecht Beutelspacher , Laura Bader +7 more

Finite geometry and combinatorics is the art of counting any phenomena that can be described by a diagram. Everyday life is full of applications; from telephones to compact disc players, … Finite geometry and combinatorics is the art of counting any phenomena that can be described by a diagram. Everyday life is full of applications; from telephones to compact disc players, from the transmission of confidential information to the codes on any item on supermarket shelves. This is a collection of thirty-five articles on covering topics such as finite projective spaces, generalized polygons, strongly regular graphs, diagram geometries and polar spaces. Included here are articles from many of the leading practitioners in the field including, for the first time, several distinguished Russian mathematicians. Many of the papers contain important new results and the growing use of computer algebra packages in this area is also demonstrated.

Open problems in finite projective spaces

2014-12-04

J. W. P. Hirschfeld , J. A. Thas

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Finite fields and Galois geometries

2008-01-01

J. A. Thas

In 1954 Segre proved the following celebrated theorem : In PG(2, q), with q odd, every oval is a nonsingular conic. Crucial for the proof is Segre's Lemma of Tangents, … In 1954 Segre proved the following celebrated theorem : In PG(2, q), with q odd, every oval is a nonsingular conic. Crucial for the proof is Segre's Lemma of Tangents, where a strong result is deduced from the simple fact that the product of the nonzero elements of GF(q) is -1. Relying on this Lemma of Tangents he was able to prove excellent theorems on certain point sets in PG(2,q). To this end he also generalized the classical theorem of Menelaus to an arbitrary collection of lines in the plane PC (2, q), no three of which are concurrent. As a corollary of these theorems good results on linear MDS codes were obtained. Here we review generalizations of the Lemma of Tangents, generalizations of Segre's generalization of the theorem of Menelaus, and applications to Hermitian curves, semiovals, circle geometries and linear MDS codes. Finally we report on recent research about generalized ovals: the elements of a generalized oval are subspaces of a projective space. To do this an appropriate 'Lemma of Tangents' type theorem is proved.

Editorial: finite geometries

2016-03-26

Dina Ghinelli , Dieter Jungnickel , Michel Lavrauw +1 more

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Finite projective spaces of three dimensions

1986-02-20

J. W. P. Hirschfeld

This self-contained and highly detailed study considers projective spaces of three dimensions over a finite field. It is the second and core volume of a three-volume treatise on finite projective … This self-contained and highly detailed study considers projective spaces of three dimensions over a finite field. It is the second and core volume of a three-volume treatise on finite projective spaces, the first volume being Projective Geometrics Over Finite Fields (OUP, 1979). The present work restricts itself to three dimensions, and considers both topics which are analogous of geometry over the complex numbers and topics that arise out of the modern theory of incidence structures. The book also examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory.

Algebraic geometry over a eld of positive characteristic

2000-01-01

Luca Giuzzi , J. W. P. Hirschfeld

Curves over nite elds not only are interesting structures in themselves, but they are also remarkable for their application to coding theory and to the study of the geometry of … Curves over nite elds not only are interesting structures in themselves, but they are also remarkable for their application to coding theory and to the study of the geometry of arcs in a nite plane. In this note, the basic properties of curves and the number of their points are recounted.

Arcs of degree four in a finite projective plane

2018-08-09

Zainab Shehab Hamed

The projective plane, PG(2;q), over a Galois field Fq is an incidence structure of points and lines. A (k;n)-arc K in PG(2;q) is a set of k points such that … The projective plane, PG(2;q), over a Galois field Fq is an incidence structure of points and lines. A (k;n)-arc K in PG(2;q) is a set of k points such that no n+1 of them are collinear but some n are collinear. A (k;n)-arc K in PG(2;q) is called complete if it is not contained in any (k+1;n)-arc. The existence of arcs for particular values of k and n pose interesting problems in finite geometry. It connects with coding theory and graph theory, with important applications in computer science. The main problem, known as the packing problem, is to determine the largest size mn(2;q) of K in PG(2;q). This problem has received much attention. Here, the work establishes complete arcs with a large number of points. In contrast, the problem to determine the smallest size tn(2;q) of a complete (k;n)-arc is mostly based on the lower bound arising from theoretical investigations. This thesis has several goals. The first goal is to classify certain (k;4)-arcs for k = 6,…,38 in PG(2;13). This classification is established through an approach in Chapter 2. This approach uses a new geometrical method; it is a combination of projective inequivalence of (k;4)-arcs up to k = 6 and certain sdinequivalent (k;4)-arcs that have sd-inequivalent classes of secant distributions for k = 7,…,38. The part related to projectively inequivalent (k;4)-arcs up to k=6 starts by fixing the frame points f1;2;3;88g and then classify the projectively inequivalent (5;4)-arcs. Among these (5;4)-arcs and (6;4)-arcs, the lexicographically least set are found. Now, the part regarding sd-inequivalent (k;4)-arcs in this method starts by choosing five sd-inequivalent (7;4)-arcs. This classification method may not produce all sd-inequivalent classes of (k;4)-arcs. However, it was necessary to employ this method due to the increasing number of (k;4)-arcs in PG(2;13) and the extreme computational difficulty of the problem. It reduces the constructed number of (k;4)-arcs in each process for large k. Consequently, it reduces the executed time for the computation which could last for years. Also, this method decreases the memory usage needed for the classification. The largest size of (k;4)-arc established through this method is k = 38. The classification of certain (k;4)-arcs up to projective equivalence, for k = 34,35,36,37,38, is also established. This classification starts from the 77 incomplete (34;4)-arcs that are constructed from the sd-inequivalent (33;4)-arcs given in Section 2.29, Table 2.35. Here, the largest size of (k;4)-arc is still k = 38. In addition, the previous process is re-iterated with a different choice of five sd-inequivalent (7;4)-arcs. The purpose of this choice is to find a new size of complete (k;4)-arc for k > 38. This particular computation of (k;4)-arcs found no complete (k;4)-arc for k > 38. In contrast, a new size of complete (k;4)-arc in PG(2;13) is discovered. This size is k = 36 which is the largest complete (k;4)-arc in this computation. This result raises the second largest size of complete (k;4)-arc found in the first classification from k = 35 to k = 36. The second goal is to discuss the incidence structure of the orbits of the groups of the projectively inequivalent (6;4)-arcs and also the incidence structures of the orbits of the groups other than the identity group of the sd-inequivalent (k;4)-arcs. In Chapter 3, these incidence structures are given for k = 6,7,8,9,10,11,12,13,14,38. Also, the pictures of the geometric configurations of the lines and the points of the orbits are described. The third goal is to find the sizes of certain sd-inequivalent complete (k;4)-arcs in PG(2;13). These sizes of complete (k;4)-arcs are given in Chapter 4 where the smallest size of complete (k;4)-arc is at most k = 24 and the largest size is at least k = 38. The fourth goal is to give an example of an associated non-singular quartic curve C for each complete (k;4)-arc and to discuss the algebraic properties of each curve in terms of the number I of inflexion points, the number jC \K j of rational points on the corresponding arc, and the number N1 of rational points of C . These curves are given in Chapter 5. Also, the algebraic properties of complete arcs of the most interesting sizes investigated in this thesis are studied. In addition, there are two examples of quartic curves C (g0 1) and C (g0 2) attaining the Hasse-Weil- Serre upper bound for the number N1 of rational points on a curve over the finite field of order thirteen. This number is 32. The fifth goal is to classify the (k;4)-arcs in PG(2;13) up to projective inequivalence for k 38 that can be obtained from these arcs. The largest size of sd-inequivalent (k;4)-arc in this process is the same as the largest size of the sd-inequivalent (k;4)-arc established in Chapter 2, that is, k = 38. In addition, the classification of (k;n)-arcs in PG(2;13) is extended from n = 4 to n = 6. This extension is given in Chapter 7 where some results of the classification of certain (k;6)-arcs for k = 9; : : : ;25 are obtained using the same method as in Chapter 2 for k = 7,…,38. This process starts by fixing a certain (8;6)-arc containing six collinear points in PG(2;13).

Projective geometry codes over prime fields

1994-01-01

J. W. P. Hirschfeld , Richard A. Shaw

Projective Planes in Algebraically Closed Fields

1991-01-01

David M. Evans , Ehud Hrushovski

We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. … We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particular configuration of four lines and six points in the geometry indicates the presence of a connected one-dimensional algebraic group.

Preface to the special issue on finite geometries

2019-01-08

Ilaria Cardinali , Michel Lavrauw , Klaus Metsch +1 more

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Book Review: Projective geometries over finite fields

1981-03-01

F. A. Sherk

It is a commonly held misconception that little research in the area of finite Desarguesian geometry is going on at present.While it is true that classical geometry as a discipline … It is a commonly held misconception that little research in the area of finite Desarguesian geometry is going on at present.While it is true that classical geometry as a discipline has been far less in vogue than it was at the beginning of this century, and that a great deal of modern research concentrates on non-Desarguesian planes, the classical case has not been ignored by all.Finite geometry in particular, inspired by the resurgence of combinatorial theory, very active work in finite algebra, and its own intrinsic appeal, has been given much attention.Classically, PG(n, K), the projective space of finite dimension n over a field K, is developed from a set of postulates, as for example in the well-known and still useful 1910 volume by Veblen and Young (Projective geometry, Blaisdell, New York).Nowadays, with linear algebra at hand and well developed, the concept can be introduced very quickly: Let V be the vector space of dimension n + 1 over K. Then PG(n 9 K) is an incidence structure consisting of subspaces of dimension m (0 < m < AI), which are simply (m + l)-dimensional subspaces of V; incidence is defined as inclusion.Subspaces of dimension 0, 1 and 2 are called points, lines and planes respectively.A subspace of maximum dimension n -1 is called a hyperplane.Characteristic properties of projective geometry are easily established; for example, when n = 2 any two distinct lines in the projective plane PG(2, K) intersect because any two distinct two-dimensional subspaces in a three-dimensional vector space share a one-dimensional subspace.One proceeds quickly to a study of collineations in PG(n, K) 9 these being nonsingular semilinear transformations in V.The introduction of sesquilinear forms into V, when applied to PG(n, AT), reproduces the classical concept of correlations (point-hyperplane correspondences), leading naturally to polarities and a study of quadric surfaces.Projective geometry received a great deal of attention in the nineteenth and early twentieth centuries, with the concentration being almost exclusively on the geometry over the real and complex fields.The problems considered were mainly such as could be handled by methods of real and complex analysis, although they were in fact often more elegantly treated by synthetic means.Thus in addition to basic properties such as the theorems of Desargues and Pappus, with their consequences, attention was centred on conies and quadric surfaces, and generalizations thereof to curves and surfaces of higher order.In real projective geometry, theorems on order and continuity were of course very important.

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Arcs in finite projective spaces

2020-01-23

Simeon Ball , Michel Lavrauw

This is an expository article detailing results concerning large arcs in finite projective spaces. It is not strictly a survey but attempts to cover the most relevant results on arcs, … This is an expository article detailing results concerning large arcs in finite projective spaces. It is not strictly a survey but attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre’s lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.

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Projective Algebraic Curves

1998-11-26

C. G. Gibson

One of the main functions of this book is to place algebraic curves in a natural setting (the complex projective plane) where they can be studied easily. For some readers, … One of the main functions of this book is to place algebraic curves in a natural setting (the complex projective plane) where they can be studied easily. For some readers, particularly those whose background is not in mathematics, this may prove to be a psychological barrier. I can only assure them that the reward is much greater than the mental effort involved. History has shown that placing algebraic curves in a natural setting provides a flood of illumination, enabling one much better to comprehend the features one meets in everyday applications.

Algebraic approach to the completeness problem for $(k,n)$-arcs in planes over finite fields

2023-01-01

Gábor Korchmáros , Gábor P. Nagy , Tamás Szőnyi

In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all … In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with $k$ close to the order of the plane, is considered a hard problem. A natural candidate to be a small $(k,n)$-arc with few characters is the set $\Omega(\mathcal{C})$ of the points of a plane curve $\mathcal{C}$ of degree $n$ (containing no linear components) such that some line meets $\mathcal{C}$ transversally in the plane, i.e. in $n$ pairwise distinct points. Let $\mathcal{C}$ be either the Hermitian curve of degree $q+1$ in $\mathrm{PG}(2,q^{2r})$ with $r\ge 1$, or the rational BKS curve of degree $q+1$ in $\mathrm{PG}(2,q^r)$ with $q$ odd and $r\ge 1$. Then $\Omega(\mathcal{C})$ has four and seven characters, respectively. Furthermore, $\Omega(\mathcal{C})$ is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that $\Omega(\mathcal{C})$ is complete for $r\ge 4$. For the rational BKS curve, $\Omega(\mathcal{C})$ is complete if and only if $r$ is even. If $r$ is odd then the uncovered points by the $(q+1)$-secants to $\Omega(\mathcal{C})$ are exactly the points in $\mathrm{PG}(2,q)$ not lying in $\Omega(\mathcal{C})$. Adding those points to $\Omega(\mathcal{C})$ produces a complete $(k,q+1)$-arc in $\mathrm{PG}(2,q^r)$, with $k=q^r+q$. The above results do not hold true for $r=2$ and there remain open the case $r=3$ for the Hermitian curve, and the cases $r=3,4$ for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for $\mathrm{PGL}(2,q)$.

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Arcs in finite projective spaces

2019-01-01

Simeon Ball , Michel Lavrauw

This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known … This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre's lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.

The Desarguesian projective plane of order eleven and related codes

2020-06-24

Awss Al-ogaidi

In a finite projective plane PG(2; q), a set K of k points is a (k; n)-arc for 2 ≤ n ≤ q - 1 if the following two properties … In a finite projective plane PG(2; q), a set K of k points is a (k; n)-arc for 2 ≤ n ≤ q - 1 if the following two properties hold: 1. Every line intersects K in at most n points. 2. There exists a line which intersects K in exactly n points. Algebraic curves of degree n give examples of (k; n)-arc; the parameter n is called the degree of the arc. In PG(2; q); the problem of finding mn(2; q) and tn(2; q) (the maximum and the minimum value of k for which a complete (k; n)-arc exists) and the problem of classifying such arcs up to projective equivalence, are crucial problems in finite geometry. One of the important application of these arcs in coding theory are projective codes that cannot be extended to larger codes. The aim of this project is to classify (k; n)-arcs if possible for 3 ≤ n ≤ 5 and to construct large arcs in PG(2; 11): Algebraic and new combinatorial methods are used to perform the classification and the construction of such arcs with different degrees. Those procedures are implemented using different open-source software packages such as GAP [35] and Orbiter [10]. We were successful in obtaining new isomorphism types of (k; 5)-arcs for k = 5,…, 13 in PG(2; 11): We have also developed a new classification algorithm for cubic curves in small projective planes. Moreover, a new upper bound is proved for the number of 5-secants of (45; 5)-arc. In addition to proving our new lower bound for the complete (k; 5)-arc in PG(2; 11): The non existence of (44; 5)-arc and (45; 5)-arc is formulated as a new conjecture for q = 11: Using an arc of degree 2 and exploiting the complement relation between arcs and blocking sets we find new 134 isomorphism types of (77; 8)-arcs in PG(2; 11):

Projective Geometry: An Introduction

2006-08-03

Rey Casse

Abstract This lucid and accessible text provides an introductory guide to projective geometry, an area of mathematics concerned with the properties and invariants of geometric figures under projection. Including numerous … Abstract This lucid and accessible text provides an introductory guide to projective geometry, an area of mathematics concerned with the properties and invariants of geometric figures under projection. Including numerous worked examples and exercises throughout, the book covers axiomatic geometry, field planes and PG(r, F), coordinating a projective plane, non-Desarguesian planes, conics and quadrics in PG(3, F). Assuming familiarity with linear algebra, elementary group theory, partial differentiation and finite fields, as well as some elementary coordinate geometry, this text is ideal for 3rd and 4th year mathematics undergraduates.

Algebraic Projective Geometry

1998-07-30

J. G. Semple , G. T. Kneebone

Abstract First published in 1952, this book has proven a valuable introduction for generations of students. It provides a clear and systematic development of projective geometry, building on concepts from … Abstract First published in 1952, this book has proven a valuable introduction for generations of students. It provides a clear and systematic development of projective geometry, building on concepts from linear algebra.

Semiarcs with long secants

2013-05-01

Bence Csajbók

Complete arcs arising from a generalization of the Hermitian curve (extended abstract)

2013-05-01

Herivelto Borges , Beatriz Motta , Fernando Torres

Linear (q+1)-fold Blocking Sets in PG(2, q4)

2000-10-01

Simeon Ball , Aart Blokhuis , Michel Lavrauw

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On defining sets for projective planes

2005-10-11

Endre Boros , Tamás Szőnyi , Krisztián Tichler

Lines, Circles, Planes and Spheres

2010-06-03

George Purdy , Justin W. Smith

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Non-existence of some 4-dimensional Griesmer codes over finite fields

2018-05-28

Kazuki Kumegawa , Tatsuya Maruta

We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$; $d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ … We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$; $d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q \ge 9$, where $g_q(4,d)=\sum_{i=0}^{3} \left\lceil d/q^i \right\rceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for $2q^3-3q^2-3q+1 \le d \le 2q^3-3q^2$, $2q^3-5q^2-2q+1 \le d \le 2q^3-5q^2$ and $q^3-q^2-rq-2 \le d \le q^3-q^2-rq$ with $4 \le r \le 6$ for $q \ge 9$ and that $n_q(4,d) \ge g_q(4,d)+1$ for $2q^3-rq^2-2q+1 \le d \le 2q^3-rq^2-q$ for $3 \le r \le (q+1)/2$, $q \ge 5$ and $2q^3-4q^2-3q+1 \le d \le 2q^3-4q^2-2q$ for $q \ge 9$, where $n_q(4,d)$ denotes the minimum length $n$ for which an $[n,4,d]_q$ code exists.

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One-factorisations of complete graphs arising from ovals in finite planes

2018-06-21

Gábor Korchmáros , Nicola Pace , Angelo Sonnino

Construction of new Griesmer codes of dimension 5

2018-11-03

Yuto Inoue , Tatsuya Maruta

On regular graphs of girth six arising from projective planes

2012-10-11

András Gács , Tamás Héger , Zsuzsa Weiner

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The Reverse construction of complete (k, n)- arcs in three-dimensional projective space PG(3,4)

2020-07-01

Aidan Essa Mustafa Sulaimaan , Nada Yassen Kasm Yahya

Abstract In this work, the complete (k, n) arcs in PG(3,4) over Galois field GF(4) can be created by removing some points from the complete arcs of degree m, where … Abstract In this work, the complete (k, n) arcs in PG(3,4) over Galois field GF(4) can be created by removing some points from the complete arcs of degree m, where m = n + 1, 3 n q2 + q is used. In addition, where k ≤ 85, we geometrically prove that the minimum complete (k, n)–arc in PG(3,4) is (5,3)-arc. A(k, n)–arcs is a set of k points no n+1 of which collinear. A(k, n)–arcs is complete unless it is embedded in an arc (k+l,n).

On the Poncelet triangle condition over finite fields

2016-12-23

Jaydeep Chipalkatti

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On a conjecture about maximum scattered subspaces of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow…

2021-09-01

Daniele Bartoli , Bence Csajbók , Maria Montanucci

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The classification of antipodal two-weight linear codes

2018-01-04

Dieter Jungnickel , Vladimir D. Tonchev

Twisted cubic and point-line incidence matrix in $$\mathrm {PG}(3,q)$$

2021-08-27

Alexander A. Davydov , Stefano Marcugini , Fernanda Pambianco

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Three new large (n, r) arcs in PG(2,31)

2020-10-11

Rumen Daskalov , Elena Metodieva

A set of n points in PG(2,q) some r of which are collinear, but every r+1 are not collinear, forms an (n, r) arc. By m r (2, … A set of n points in PG(2,q) some r of which are collinear, but every r+1 are not collinear, forms an (n, r) arc. By m r (2, q) the maximum size of a such arc is denoted. In this talk (n, r) arcs with parameters (699, 24), (769, 26), and (838, 28) in PG(2,31) will be presented. The constructed arcs improve the respective lower bounds on m r (2, 31) in [8]. As a consequence there exist three new three-dimensional linear codes over GF(31).

On homogeneous planar functions

2014-11-07

Tao Feng

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Partitions of the lines in PG<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e625" altimg="si10.svg"><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> into multifold spreads for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e645" altimg="si11.svg"><mml:mrow><…

2020-03-02

Miwako Mishima , Nobuko Miyamoto , Masakazu Jimbo

NONBINARY DELSARTE–GOETHALS CODES AND FINITE SEMIFIELDS

2020-05-07

I. F. Rúa

Abstract Symplectic finite semifields can be used to construct nonlinear binary codes of Kerdock type (i.e., with the same parameters of the Kerdock codes, a subclass of Delsarte–Goethals codes). In … Abstract Symplectic finite semifields can be used to construct nonlinear binary codes of Kerdock type (i.e., with the same parameters of the Kerdock codes, a subclass of Delsarte–Goethals codes). In this paper, we introduce nonbinary Delsarte–Goethals codes of parameters $(q^{m+1}\ ,\ q^{m(r+2)+2}\ ,\ {\frac{q-1}{q}(q^{m+1}-q^{\frac{m+1}{2}+r})})$ over a Galois field of order $q=2^l$ , for all $0\le r\le\frac{m-1}{2}$ , with m ≥ 3 odd, and show the connection of this construction to finite semifields.

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Small complete caps in PG(4n+1,q)${\rm PG}(4n+1, q)$

2022-11-05

Antonio Cossidente , Bence Csajbók , Giuseppe Marino +1 more

In this paper we prove the existence of a complete cap of PG ( 4 n + 1 , q ) ${\rm PG}(4n+1, q)$ of size 2 ( q 2 … In this paper we prove the existence of a complete cap of PG ( 4 n + 1 , q ) ${\rm PG}(4n+1, q)$ of size 2 ( q 2 n + 1 − 1 ) / ( q − 1 ) $2(q^{2n+1}-1)/(q-1)$ , for each prime power q > 2 $q>2$ . It is obtained by projecting two disjoint Veronese varieties of PG ( 2 n 2 + 3 n , q ) ${\rm PG}(2n^2+3n, q)$ from a suitable ( 2 n 2 − n − 2 ) $(2n^2-n-2)$ -dimensional projective space. This shows that the trivial lower bound for the size of the smallest complete cap of PG ( 4 n + 1 , q ) ${\rm PG}(4n+1, q)$ is essentially sharp.

Inherited conics in Hall planes

2019-01-11

Aart Blokhuis , István Kovács , Gábor P. Nagy +1 more

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Parallelisms of $$\textrm{PG}(3,4)$$ with a Great Number of Regular Spreads

2024-01-01

Svetlana Topalova , Stela Zhelezova

Hearing shapes of drums: Mathematical and physical aspects of isospectrality

2010-08-06

Olivier Giraud , Koen Thas

In a celebrated paper ``Can one hear the shape of a drum?'' M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards … In a celebrated paper ``Can one hear the shape of a drum?'' M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.

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On Cubic Surfaces Over a Field of Characteristic 3

1983-01-01

Massimo de Finis , M.J. de Resmin

Factor-Group-Generated Polar Spaces and (Multi-)Qudits

2009-10-13

Hans Havlicek

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries.Here, we succeeded in finding a general unifying framework for all these relations.We introduce … Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries.Here, we succeeded in finding a general unifying framework for all these relations.We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group G, we first construct vector spaces over GF(p), p a prime, by factorising G over appropriate normal subgroups.Then, by expressing GF(p) in terms of the commutator subgroup of G, we construct alternating bilinear forms, which reflect whether or not two elements of G commute.Restricting to p = 2, we search for "refinements" in terms of quadratic forms, which capture the fact whether or not the order of an element of G is ≤ 2. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a "condensation" of several distinct elements of G. Finally, several well-known physical examples (single-and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.

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The moment map of a Lie group representation

1992-01-01

N. J. Wildberger

Given an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m times m"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>×</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m \times m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Hadamard matrix one can extract <inline-formula … Given an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m times m"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>×</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m \times m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Hadamard matrix one can extract <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{m^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> symmetric designs on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m minus 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points each of which extends uniquely to a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-design. Further, when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a square, certain Hadamard matrices yield symmetric designs on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary association of linear codes with designs. This leads naturally to the notion, defined for any prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivalence for Hadamard matrices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="24 times 24"> <mml:semantics> <mml:mrow> <mml:mn>24</mml:mn> <mml:mo>×</mml:mo> <mml:mn>24</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">24 \times 24</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices fall into only six <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivalence classes. In the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="16 times 16"> <mml:semantics> <mml:mrow> <mml:mn>16</mml:mn> <mml:mo>×</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">16 \times 16</mml:annotation> </mml:semantics> </mml:math> </inline-formula> case, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivalence is identical to the standard equivalence, but our results illuminate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group, why two of the matrices cannot be obtained from a symmetric design on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="16"> <mml:semantics> <mml:mn>16</mml:mn> <mml:annotation encoding="application/x-tex">16</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.

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Minimal complete arcs in PG(2,q), q<= 32

2010-05-19

Stefano Marcugini , Alfredo Milani , Fernanda Pambianco

In this paper it has been verified, by a computer-based proof, that the smallest size of a complete arc is 14 in PG(2,31) and in PG(2,32). Some examples of such … In this paper it has been verified, by a computer-based proof, that the smallest size of a complete arc is 14 in PG(2,31) and in PG(2,32). Some examples of such arcs are also described.

From a projective invariant to some new properties of algebraic hypersurfaces

2014-07-14

Zhongxuan Luo , Xinchen Zhou , Xianfeng Gu

A new lower bound for the smallest complete (k, n)-arc in $$\mathrm {PG}(2,q)$$ PG ( 2 , q )

2018-12-21

Salam Abdulqader Falih Alabdullah , J. W. P. Hirschfeld

In $$\mathrm {PG}(2,q)$$ , the projective plane over the field $$\mathbf{F}_{q}$$ of q elements, a (k, n)-arc is a set $$\mathcal {K}$$ of k points with at most n points … In $$\mathrm {PG}(2,q)$$ , the projective plane over the field $$\mathbf{F}_{q}$$ of q elements, a (k, n)-arc is a set $$\mathcal {K}$$ of k points with at most n points on any line of the plane. A fundamental question is to determine the values of k for which $$\mathcal {K}$$ is complete, that is, not contained in a $$(k+1,n)$$ -arc. In particular, what are the smallest and largest values of k for a complete $$\mathcal {K}$$ , denoted by $$t_n(2,q)$$ and $$m_n(2,q)$$ ? Here, a new lower bound for $$t_n(2,q)$$ is established and compared to known values for small q.

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Weierstrass points on Kummer extensions

2018-07-20

Miriam Abdón , Herivelto Borges , Luciane Quoos

Abstract For Kummer extensions given by y m = f ( x ), we discuss conditions for an integer to be a Weierstrass gap at a place P . In … Abstract For Kummer extensions given by y m = f ( x ), we discuss conditions for an integer to be a Weierstrass gap at a place P . In the case of fully ramified places, the conditions are necessary and sufficient. As a consequence, we extend independent results of several authors. Moreover, we show that if the Kummer extension is 𝔽 q 2 -maximal and f ( x ) ∈ 𝔽 q 2 [ x ] has at least two roots with the same multiplicity λ coprime to m , then m divides 2( q + 1). Under the extra condition that either m or the multiplicity of a third root of f ( x ) is odd, we conclude that m divides q + 1.

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None

2002-01-01

Marialuisa J. de Resmini , Dina Ghinelli , Dieter Jungnickel

On the equivalence of certain quasi‐Hermitian varieties

2022-12-07

Angela Aguglia , Luca Giuzzi

In [A. Aguglia, A. Cossidente, G. Korchmaros, "On quasi-Hermitian varieties", J. Comb. Des. 20 (2012), 433-447] new quasi-Hermitian varieties ${\mathcal M}_{\alpha,\beta}$ in $\mathrm{PG}(r,q^2)$ depending on a pair of parameters $\alpha,\beta$ … In [A. Aguglia, A. Cossidente, G. Korchmaros, "On quasi-Hermitian varieties", J. Comb. Des. 20 (2012), 433-447] new quasi-Hermitian varieties ${\mathcal M}_{\alpha,\beta}$ in $\mathrm{PG}(r,q^2)$ depending on a pair of parameters $\alpha,\beta$ from the underlying field $\mathrm{GF}(q^2)$ have been constructed. In the present paper we determine the projective equivalence classes of such varieties for $r=3$ and $q$ odd.

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Discovering Doily in PG(2,5)

2023-04-17

Stefano Innamorati

W. L. Edge proved that the internal points of a conic in PG(2,5), together with the collinear triples on the non-secant lines, form the Desargues configuration. M. Saniga showed an … W. L. Edge proved that the internal points of a conic in PG(2,5), together with the collinear triples on the non-secant lines, form the Desargues configuration. M. Saniga showed an intimate connection between Desargues configurations and the generalized quadrangles of order two, GQ(2,2), whose representation has been dubbed “the doily” by Stan Payne in 1973. In this paper we prove that the external points of a conic in PG(2,5), together with the collinear and non-collinear triples on the non-tangent lines, form the generalized quadrangle of order two.

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Quantum systems with finite Hilbert space: Galois fields in quantum mechanics

2007-08-01

A. Vourdas

A 'Galois quantum system' in which the position and momentum take values in the Galois field GF(pℓ) is considered. It is comprised of ℓ-component systems which are coupled in a … A 'Galois quantum system' in which the position and momentum take values in the Galois field GF(pℓ) is considered. It is comprised of ℓ-component systems which are coupled in a particular way and is described by a certain class of Hamiltonians. Displacements in the GF(pℓ) × GF(pℓ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF(pℓ)) group. Wigner and Weyl functions are defined and their properties are studied. Frobenius symmetries, which are based on Frobenius automorphisms in the theory of Galois fields, are a unique feature of these systems (for ℓ ⩾ 2). If they commute with the Hamiltonian, there are constants of motion which are discussed. An analytic representation in the ℓ-sheeted complex plane provides an elegant formalism that embodies the properties of Frobenius transformations. The difference between a Galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed.

Ovals in Desarguesian planes.

1990-01-01

Christine M. O’Keefe

Minimum distance of Hermitian two-point codes

2010-01-08

Seungkook Park

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On zeros of Kloosterman sums

2011-01-03

Petr Lisoněk , Marko Moisio

None

2002-01-01

Massimo Giulietti , Fernanda Pambianco , Fernando Torres +1 more

Quotient curves of the Suzuki curve

2006-01-01

Massimo Giulietti , Gábor Korchmáros , Fernando Torres

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Caps in PG(n, q),q even,n?3

1993-02-01

Leo Storme , T. Sz�nyi

The complete k-arcs of PG(2, 27) and PG(2, 29)

2010-07-29

Kris Coolsaet , Heide Sticker

A full classification (up to equivalence) of all complete k-arcs in the Desarguesian projective planes of order 27 and 29 was obtained by computer.The resulting numbers of complete arcs are … A full classification (up to equivalence) of all complete k-arcs in the Desarguesian projective planes of order 27 and 29 was obtained by computer.The resulting numbers of complete arcs are tabulated according to size of the arc and type of the automorphism group, and also according to the type of algebraic curve into which they can be embedded.For the arcs with the larger automorphism groups, explicit descriptions are given.The algorithm used for generating the arcs is an application of isomorphfree backtracking using canonical augmentation, an adaptation of an earlier algorithm by the authors.Part of the computer results can be generalized to other values of q: two families of arcs are presented (of size 12 and size 18) for which the symmetric group S 4 is a group of automorphisms.

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