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 …
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.
This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many …
This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
Studies a class of projective systems and linear codes corresponding to Hermitian varieties over finite fields. The weight hierarchy, also known as the set of generalized Hamming weights, of the …
Studies a class of projective systems and linear codes corresponding to Hermitian varieties over finite fields. The weight hierarchy, also known as the set of generalized Hamming weights, of the code is calculated. The higher weight distribution is also found.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
The British Combinatorial Conference is held every two years and is now a key event for mathematicians worldwide, working in combinatorics. This volume is published on the occasion of the …
The British Combinatorial Conference is held every two years and is now a key event for mathematicians worldwide, working in combinatorics. This volume is published on the occasion of the 18th meeting, which was held 1st-6th July 2001 at the University of Sussex. The papers contained here are surveys contributed by the invited speakers, and are thus of a quality befitting the event. There is also a tribute to Crispin Nash-Williams, past chairman of the British Combinatorial Committee. The diversity of the subjects covered means that this will be a valuable reference for researchers in combinatorics. However, graduate students will also find much here that could be of use for stimulating future research.
Abstract In a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a …
Abstract In a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.
The known results on the maximum size of an arc in a projective space or equivalently the maximum length of a maximum distance separable linear code are surveyed. It is …
The known results on the maximum size of an arc in a projective space or equivalently the maximum length of a maximum distance separable linear code are surveyed. It is then shown that this maximum is q+1 for all dimensions up to q in the cases that q=11 and q=13; the result for q=11 was previously known. The strategy is to first show that a 11-arc in PG (3,11) and a 12-arc in PG (3,13) are subsets of a twisted cubic, that is, a normal rational curve.
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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.
The object of this paper is to consider the existence of the double-six over GF (2 n ) and particularly over GF (4).
The object of this paper is to consider the existence of the double-six over GF (2 n ) and particularly over GF (4).
In 1849, Cayley and Salmon discovered that a general cubic surface in projective space of three dimensions over the complex numbers has twenty-seven lines on it. They remarked that all …
In 1849, Cayley and Salmon discovered that a general cubic surface in projective space of three dimensions over the complex numbers has twenty-seven lines on it. They remarked that all the properties of the twenty-seven lines would not become apparent until a better notation than they had given was found. This notation was discovered by Schläfli in 1858 in the double-six theorem (henceforth referred to as given five skew lines a 1 , …, a 5 with a single transversal b 6 such that no four of the a i lie in a regulus, the four a i excluding a j have a second transversal b j and the five lines b 1 , …, b 5 thus obtained have a transversal a 6 — the completing line of the double-six . The other fifteen lines of the cubic surface are then , where a i b j is the plane containing a i and b j .
By Branko Grünbaum: 399 pp., US$75.00, isbn 978-0-8218-4308-6 (American Mathematical Society, Providence, RI, 2009).
By Branko Grünbaum: 399 pp., US$75.00, isbn 978-0-8218-4308-6 (American Mathematical Society, Providence, RI, 2009).
The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely …
The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely irreducible polynomial with coefficients in a finite field Fq.
In Hirschfeld (J Austral Math Soc 4(1):83–89, 1964), the existence of the cubic surface which arises from a double-six over the finite field of order four was considered. In Hirschfeld …
In Hirschfeld (J Austral Math Soc 4(1):83–89, 1964), the existence of the cubic surface which arises from a double-six over the finite field of order four was considered. In Hirschfeld (Rend Mat Appl 26:115–152, 1967), the existence and the properties of the cubic surfaces over the finite fields of odd and even order was discussed and classified over the fields of order seven, eight, nine. In this paper, cubic surfaces with twenty-seven lines over the finite field of thirteen elements are classified.
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.
A lower bound for the size of a complete cap of the polar space H(n, q 2 ) associated to the non-degenerate Hermitian variety U n is given; this turns …
A lower bound for the size of a complete cap of the polar space H(n, q 2 ) associated to the non-degenerate Hermitian variety U n is given; this turns out to be sharp for even q when n = 3.Also, a family of caps of H(n, q 2 ) is constructed from F q 2 -maximal curves.Such caps are complete for q even, but not necessarily for q odd.
Let F be a cubic surface with 27 lines in PG(3,q). Theorem 30.1 in Manin [2] states that, if q > 34, then there exists a point of F on …
Let F be a cubic surface with 27 lines in PG(3,q). Theorem 30.1 in Manin [2] states that, if q > 34, then there exists a point of F on none of its lines. There is, however, sufficient information in [1] to work out the precise list of cubic surfaces with no such point.
How many points are there on a curve with coordinates in a given finite field when the curve has (a) no singular points or (b) singular points counted once or …
How many points are there on a curve with coordinates in a given finite field when the curve has (a) no singular points or (b) singular points counted once or (c) singular points counted with multiplicity?
Preface Table of contents Introduction L. Bader: Flocks of cones and generalized hexagons L.M. Batten and J.M.N. Brown: Group partitions, affine spaces and translation groups A. Blokhuis and F. Mazzocca: …
Preface Table of contents Introduction L. Bader: Flocks of cones and generalized hexagons L.M. Batten and J.M.N. Brown: Group partitions, affine spaces and translation groups A. Blokhuis and F. Mazzocca: On maximal sets of nuclei in PG(2,q) and quasi-odd sets in AG(2,q) A. Blokhuis, A.R. Calderbank, K. Metsch and A. Seress: An embedding theorem for partial linear spaces A.E. Brouwer: A non-degenerate generalized quadrangle with lines of size four is finite J.M.N. Brown: On abelian collineation groups of finite projective planes F. Buekenhout: More geometry for Hering's 3 :SL(2,13) A.M. Cohen: Presentations for certain finite quaternionic relection groups B.N. Cooperstein and E.E. Shult: Geometric hyperplanes of embeddable Lie incidence geometries F. De Clerck, A. Del Fra, and D. Ghinelli: Pointsets in partial geometries A. Del Fra, D. Ghinelli, and A. Pasini: One diagram for many geometries M.J. de Resmini: On an exceptional semi-translation plane W.HL Haemers, D.G. Higman, and S.A. Hobart: Strongly regular graphs induced by polarities of symmetric designs J.W.P. Hirschfeld and Tzonyi: A Problem on squares in a finite field and its application to geometry C. Ho and A. Goncalves: On totally irregular simple collineation groups D.R. Hughes: On some rank 3 partial geometries D. Jungnickel and V.D. Tonchev: Intersection numbers of quasi-multiples of symmetric designs H. kaneta and T. Maruta: The discriminant of a cubic curve W.M. Kantor: Automorphism groups of some generalized quadrangles G. Korchmaros: Collineation groups of (q + t, t)-arcs of type (0,2,t) in a Desarguesian plane of order q Y. Liu and Z. Wan: Pseudo-symplectic geometries over finite fields of characteristic two S. Lowe: Solutions of quadratic equations in a groupring and partial addition sets G. Lunardon: A remark on the derivation of flocks T. Maruta: A geometric approach to semi-cyclic codes G. Mason: Remarks on non-associative Galois theory G.E. Moorhouse: Codes of nets with translations C.M. O'Keefe, T Penttila, and C.M. Praeger: Stabilizers of hyperovals in PG(2, 32) J. Siemons and B. Webb: On a problem of Wielandt and a question by Dembowski E. Spence: (40, 13,4)-designs derived from strongly regular graphs L. Storme and J.A. Thas: Generalized Reed-Solomon codes and normal rational curves: an improvement of results by Seroussi and Roth H. Van Maldeghem: Common characterizations of the finite Moufang polygons J.F. Voloch: Complete arcs in Galois planes of non-square order F. Wettl: Internal nuclei of k-sets in finite projective spaces of three dimensions P.R. Wild: Menon difference sets and relative difference sets List of Talks List of Participants.
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 …
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod 4). Here, a maximal curve with genus g_2 and a non-singular plane model is characterized as a Fermat curve of degree (r+1)/2.
Arcs and caps are fundamental structures in finite projective spaces. They can be generalised. Here, a survey is given of some important results on these objects, in particular on generalised …
Arcs and caps are fundamental structures in finite projective spaces. They can be generalised. Here, a survey is given of some important results on these objects, in particular on generalised ovals and generalised ovoids. This paper also contains recent results and several open problems.
Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how …
Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadratic forms. The Arf invariant, in particular, plays a significant role.
Abstract In this paper, the classification of ( k ;4)-arcs up to projective inequivalence for k < 10 in PG(2,13) is introduced in details according to their inequivalent number, stabilisers, …
Abstract In this paper, the classification of ( k ;4)-arcs up to projective inequivalence for k < 10 in PG(2,13) is introduced in details according to their inequivalent number, stabilisers, the action of each stabiliser on the associated arc, and the inequivalent classes N c of secant distributions of arcs. Here, the strategy is to start from the projective line PG(1,13) where there are three projectively inequivalent tetrads.
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.
In Hirschfeld (J Austral Math Soc 4(1):83–89, 1964), the existence of the cubic surface which arises from a double-six over the finite field of order four was considered. In Hirschfeld …
In Hirschfeld (J Austral Math Soc 4(1):83–89, 1964), the existence of the cubic surface which arises from a double-six over the finite field of order four was considered. In Hirschfeld (Rend Mat Appl 26:115–152, 1967), the existence and the properties of the cubic surfaces over the finite fields of odd and even order was discussed and classified over the fields of order seven, eight, nine. In this paper, cubic surfaces with twenty-seven lines over the finite field of thirteen elements are classified.
By Branko Grünbaum: 399 pp., US$75.00, isbn 978-0-8218-4308-6 (American Mathematical Society, Providence, RI, 2009).
By Branko Grünbaum: 399 pp., US$75.00, isbn 978-0-8218-4308-6 (American Mathematical Society, Providence, RI, 2009).
This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many …
This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
On the Semiprimitivity of Cyclic Codes (Y Aubry & P Langevin) An Optimal Unramified Tower of Function Fields (K Brander) Galois Invariant Smoothness Basis (J-M Couveignes) Decoding of Scroll Codes …
On the Semiprimitivity of Cyclic Codes (Y Aubry & P Langevin) An Optimal Unramified Tower of Function Fields (K Brander) Galois Invariant Smoothness Basis (J-M Couveignes) Decoding of Scroll Codes (G Hitching & T Johnsen) Fuzzy Pairing-based CL-PKC (M Kiviharju) On Quadratic Extensions of Cyclic Projective Planes (H F Law & P Wong) On the Number of Boolean Resilient Functions (S Mesnager) Symmetric Cryptography and Algebraic Curves (F Voloch) Partitions of Vector Spaces over Finite Fields (Y Zelenyuk) and other papers.
How many points are there on a curve with coordinates in a given finite field when the curve has (a) no singular points or (b) singular points counted once or …
How many points are there on a curve with coordinates in a given finite field when the curve has (a) no singular points or (b) singular points counted once or (c) singular points counted with multiplicity?
A lower bound for the size of a complete cap of the polar space H(n, q 2 ) associated to the non-degenerate Hermitian variety U n is given; this turns …
A lower bound for the size of a complete cap of the polar space H(n, q 2 ) associated to the non-degenerate Hermitian variety U n is given; this turns out to be sharp for even q when n = 3.Also, a family of caps of H(n, q 2 ) is constructed from F q 2 -maximal curves.Such caps are complete for q even, but not necessarily for q odd.
The British Combinatorial Conference is held every two years and is now a key event for mathematicians worldwide, working in combinatorics. This volume is published on the occasion of the …
The British Combinatorial Conference is held every two years and is now a key event for mathematicians worldwide, working in combinatorics. This volume is published on the occasion of the 18th meeting, which was held 1st-6th July 2001 at the University of Sussex. The papers contained here are surveys contributed by the invited speakers, and are thus of a quality befitting the event. There is also a tribute to Crispin Nash-Williams, past chairman of the British Combinatorial Committee. The diversity of the subjects covered means that this will be a valuable reference for researchers in combinatorics. However, graduate students will also find much here that could be of use for stimulating future research.
The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely …
The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely irreducible polynomial with coefficients in a finite field Fq.
By Oliver Pretzel: 192 pp., £35.00, isbn 0 19 850039 4 (Clarendon Press, 1998).
By Oliver Pretzel: 192 pp., £35.00, isbn 0 19 850039 4 (Clarendon Press, 1998).
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 …
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod 4). Here, a maximal curve with genus g_2 and a non-singular plane model is characterized as a Fermat curve of degree (r+1)/2.
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 …
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.
Abstract Let pc(r)(n,q) be the set of r-spaces of PG(n,q) and let ø (r;n,q) be the cardinality of PG(r)(n,q). Also, let 𝒳 (s,r; n,q) be the number of r-spaces through …
Abstract Let pc(r)(n,q) be the set of r-spaces of PG(n,q) and let ø (r;n,q) be the cardinality of PG(r)(n,q). Also, let 𝒳 (s,r; n,q) be the number of r-spaces through r-space of PG(n,q).
Abstract The objective of this section is to associate an algebraic curve to an arc of degree two. This is, perhaps, the most significant result in the book.
Abstract The objective of this section is to associate an algebraic curve to an arc of degree two. This is, perhaps, the most significant result in the book.
Abstract For any m = −1,0,1,2,... ,n, a subspace of dimension m, or m-space, of PG(n, K) is a set of points all of whose representing vectors form, together with …
Abstract For any m = −1,0,1,2,... ,n, a subspace of dimension m, or m-space, of PG(n, K) is a set of points all of whose representing vectors form, together with the zero, a subspace of dimension m+1 of V = V(n 4+1, q); it is denoted by Πm A subspace of dimension zero has already been called a point; a subspace of dimension −1 is the empty set. Subspaces of dimension one, two, three are respectively a line, a plane, a solid. A subspace of dimension n −1 is a prime or hyperplane-, a subspace of dimension n−2 is a secundum. A subspace of dimension n −r is also referred to as a subspace of codimension r. The set of m-spaces is denoted PG(m)(n,q).
Abstract A spread ℱ of PG(n, q) by r-spaces is a set of r-spaces which partitions PG(n, q); that is, every point of PG(n, q) lies in exactly one r-space …
Abstract A spread ℱ of PG(n, q) by r-spaces is a set of r-spaces which partitions PG(n, q); that is, every point of PG(n, q) lies in exactly one r-space of ℱ. Hence any two r-spaces of ℱ are disjoint. Here it is convenient to write θ (n, q) for ϕ (0;n, q).
Abstract Although a superficial examination indicates that characteristic two is a special case, the results do not bear this out; however, slightly different methods are required to obtain the results. …
Abstract Although a superficial examination indicates that characteristic two is a special case, the results do not bear this out; however, slightly different methods are required to obtain the results. Apart from the value of q modulo 3, the value modulo 4 turns out to be relevant. In fact, the value of q modulo 12 provides the separation of prime powers needed; this is given in Table 11.1.
Abstract In each case, the form and the variety are degenerate if there is a change of coordinate system which reduces the form to one in fewer variables; otherwise, the …
Abstract In each case, the form and the variety are degenerate if there is a change of coordinate system which reduces the form to one in fewer variables; otherwise, the form and the variety are non-degenerate.
Abstract A blocking set is minimal or irreducible if B\{P} is not a blocking set for every P in B. A blocking set of Redei type is a blocking set …
Abstract A blocking set is minimal or irreducible if B\{P} is not a blocking set for every P in B. A blocking set of Redei type is a blocking set of size q + m that has an m-secant.
Abstract A (k; n)-arc of a projective plane is the complement of a t-fold blocking set with n + t = q + 1. For this reason many of the …
Abstract A (k; n)-arc of a projective plane is the complement of a t-fold blocking set with n + t = q + 1. For this reason many of the results occurring in the next chapter in relation to multiple blocking sets may be relevant to (k; n)-arcs. In general, (k; n)-arcs are considered when n is small and t-fold blocking sets when t is small.
Abstract 1.1 Definitions and existence (i) A field is a set K closed under two operations+, × such that (a) (K, +) is an abelian group with identity O; (b) …
Abstract 1.1 Definitions and existence (i) A field is a set K closed under two operations+, × such that (a) (K, +) is an abelian group with identity O; (b) (Ko, ×) is an abelian group with identity 1, where Ko= K\{O); (c) x(y + z) = xy + xz, (x + y)z = xz + yz, for all .r,y, z in K. (ii) A finite field is a field with only a finite number of elements. (iii) The characteristic of a finite field K is the smallest positive integer (and hence a prime) p such that px = 0 for all x in K.
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 …
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.
By J. W. P. Hirschfeld and J. A. Thas: 407 pp., £55.00, ISBN 0 19 853537 6 (Clarendon Press, 1991).
By J. W. P. Hirschfeld and J. A. Thas: 407 pp., £55.00, ISBN 0 19 853537 6 (Clarendon Press, 1991).
For any projective embedding of a non-singular irreducible complete algebraic curve defined over a finite field, we obtain an upper bound for the number of its rational points. The constants …
For any projective embedding of a non-singular irreducible complete algebraic curve defined over a finite field, we obtain an upper bound for the number of its rational points. The constants in the bound are related to the Weierstrass order-sequence associated with the projective embedding. The bounds obtained lead to a proof of the Riemann hypothesis for curves over finite fields and yield several improvements on it.
By J. W. P. Hirschfeld: 555 pp., £65.00, isbn 0 19 850295 8 (Clarendon Press, 1998).
By J. W. P. Hirschfeld: 555 pp., £65.00, isbn 0 19 850295 8 (Clarendon Press, 1998).
1. Let be a finite projective plane (8, §17), i.e. a projective space of dimension 2 over a Galois field γ. We suppose that γ has characteristic p ≠ 2, …
1. Let be a finite projective plane (8, §17), i.e. a projective space of dimension 2 over a Galois field γ. We suppose that γ has characteristic p ≠ 2, hence order q = p n , where p is an odd prime and h is a positive integer. It is well known that every straight line and every non-singular conic of then contains q + 1 points exactly.
An M × N matrix is associated with each ordered k-arc in a finite projective space of order N (k = M + N + 2.) The matrix is a …
An M × N matrix is associated with each ordered k-arc in a finite projective space of order N (k = M + N + 2.) The matrix is a projective invariant for ordered k-arcs in the space. The set of these matrices is denoted by Ω. Elements of the symmetric group Sk act on ordered arcs by permuting points. This induces a definition of Sk as a group of operators on Ω, whose orbits correspond to projectively distinct unordered k-arcs. Application of theorems of Burnside and Cauchy leads to results concerning the number of orbits of k-arcs in PG(N, q) under projectivity and under collineation. A subset of Ω is defined which contains representatives of each orbit under Sk. The reduced set of “normal” matrices is used by a counting algorithm. The results of this paper are applied to counting the projectively distinct unordered k-arcs (all k) in PG(2, 11) and PG(2, 13).
Abstract A fundamental quantity in the geometry of projective spaces over finite fields is the cardinality of the second largest complete arc in PG(2, q) denoted by m’(2, q).
Abstract A fundamental quantity in the geometry of projective spaces over finite fields is the cardinality of the second largest complete arc in PG(2, q) denoted by m’(2, q).
Let q be the finite field of q elements. Denote by S r q the projective space of dimension r over q . In S r,q , where r ≥ …
Let q be the finite field of q elements. Denote by S r q the projective space of dimension r over q . In S r,q , where r ≥ 2, a k -arc is defined (see [4]) as a set of k points such that no j + 2 lie in a S j,q , for j = 1,2,…, r−1. (For a k -arc with k > r, this last condition holds for all j when it holds for j = r −1.) A rational curve C n of order n in S r,q , is the set
The known results on the maximum size of an arc in a projective space or equivalently the maximum length of a maximum distance separable linear code are surveyed. It is …
The known results on the maximum size of an arc in a projective space or equivalently the maximum length of a maximum distance separable linear code are surveyed. It is then shown that this maximum is q+1 for all dimensions up to q in the cases that q=11 and q=13; the result for q=11 was previously known. The strategy is to first show that a 11-arc in PG (3,11) and a 12-arc in PG (3,13) are subsets of a twisted cubic, that is, a normal rational curve.
Throughout this paper q = 2 h with h ≥ 4, and PG ( r, q ) stands for the r -dimensional projective space over the finite field GF ( …
Throughout this paper q = 2 h with h ≥ 4, and PG ( r, q ) stands for the r -dimensional projective space over the finite field GF ( q ) with q elements.