Skip to main content
×
×
Home
Finite Geometry and Combinatorial Applications
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 3
  • Cited by
    This (lowercase (translateProductType product.productType)) has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Hua, Michael Damelin, Steven B. Sun, Jeffrey and Yu, Mingchao 2018. The truncated and supplemented Pascal matrix and applications. Involve, a Journal of Mathematics, Vol. 11, Issue. 2, p. 243.


    Ball, Simeon 2017. Extending small arcs to large arcs. European Journal of Mathematics,


    Ball, Simeon and De Beule, Jan 2017. On Subsets of the Normal Rational Curve. IEEE Transactions on Information Theory, Vol. 63, Issue. 6, p. 3658.


    ×
  • Simeon Ball, Universitat Politècnica de Catalunya, Barcelona

  • Export citation
  • Recommend to librarian
  • Recommend this book

    Email your librarian or administrator to recommend adding this book to your organisation's collection.

    Finite Geometry and Combinatorial Applications
    • Online ISBN: 9781316257449
    • Book DOI: https://doi.org/10.1017/CBO9781316257449
    Please enter your name
    Please enter a valid email address
    Who would you like to send this to *
    ×
  • Buy the print book

Book description

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.

Refine List
Actions for selected content:
Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive
  • Send content to

    To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to .

    To send content items to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

    Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

    Find out more about the Kindle Personal Document Service.

    Please be advised that item(s) you selected are not available.
    You are about to send
    ×

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
×
References
Albert A. A. (1960), Finite division algebras and finite planes, in Proc. Sympos. Appl. Math., Vol. 10, American Mathematical Society, Providence, R.I., pp. 53–70.
Albert A. A. & Sandler R. (1968), An Introduction to Finite Projective Planes, Holt, Rinehart and Winston, New York-Toronto, Ont.-London.
Alderson T. L. & Gács A. (2009), ‘On the maximality of linear codes’, Des. Codes Cryptogr. 53(1), 59–68.
Alon N., Rónyai L. & Szabó T. (1999), ‘Norm-graphs: variations and applications’, J. Combin. Theory Ser. B 76(2), 280–290.
André J. (1954), ‘Uber nicht-Desarguessche Ebenen mit transitiver Translations- gruppe’, Math. Z. 60, 156–186.
Atiyah M. F. & Macdonald I. G. (1969), Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.
Ball S. (2003), ‘The number of directions determined by a function over a finite field’, J. Combin. Theory Ser. A 104(2), 341–350.
Ball S. (2012), ‘On sets of vectors of a finite vector space in which every subset of basis size is a basis’, J. Eur. Math. Soc. (JEMS) 14(3), 733–748.
Ball S. & Blokhuis A. (1998), ‘An easier proof of the maximal arcs conjecture’, Proc. Amer. Math. Soc. 126(11), 3377–3380.
Ball S., Blokhuis A. & Mazzocca F. (1997), ‘Maximal arcs in Desarguesian planes of odd order do not exist’, Combinatorica 17(1), 31–41.
Ball S. & De Beule J. (2012), ‘On sets of vectors of a finite vector space in which every subset of basis size is a basis II’, Des. Codes Cryptogr. 65(1–2), 5–14.
Ball S., Ebert G. & Lavrauw M. (2007), ‘A geometric construction of finite semi-fields’, J. Algebra 311(1), 117–129.
Ball S. & Gács A. (2009), ‘On the graph of a function over a prime field whose small powers have bounded degree’, European J. Combin. 30(7), 1575–1584.
Ball S. & Pepe V. (2012), ‘Asymptotic improvements to the lower bound of certain bipartite Turan numbers’, Combin. Probab. Comput. 21(3), 323–329.
Ball S. & Zieve M. (2004), Symplectic spreads and permutation polynomials, in Finite Fields and Applications, Vol. 2948 of Lecture Notes in Comput. Sci., Springer, Berlin pp. 79–88.
Barlotti A. (1955), ‘Un'estensione del teoremadi Segre-Kustaanheimo’, Boll. Un. Mat. Ital. (3) 10, 498–506.
Barwick S. & Ebert G. (2008), Unitals in Projective Planes, Springer Monographs in Mathematics, Springer, New York.
Bennett M. K. (1995), Affine and Projective Geometry, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.
Berlekamp E. R. (1968), Algebraic Coding Theory, McGraw-Hill Book Co., New York-Toronto|Ont.-London.
Beth T., Jungnickel D. & Lenz H. (1999a), Design Theory. Vol. I, Vol.69 of Encyclopedia of Mathematics and its Applications, second edn, Cambridge University Press, Cambridge.
Beth T., Jungnickel D. & Lenz H. (1999b), Design theory. Vol. II, Vol.78 of Encyclopedia of Mathematics and its Applications, second edn, Cambridge University Press, Cambridge.
Betten A., Braun M., Fripertinger H., Kerber A., Kohnert A. & Wassermann A. (2006), Error-Correcting Linear Codes, Vol. 18 of Algorithms and Computation in Mathematics, Springer-Verlag, Berlin.
Beutelspacher A. & Rosenbaum U. (1998), Projective Geometry: From Foundations to Applications, Cambridge University Press, Cambridge.
Bierbrauer J. (2005), Introduction to Coding Theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL.
Bierbrauer J. (2009), New commutative semifields and their nuclei, in Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Vol. 5527 of Lecture Notes in Comput. Sci., Springer, Berlin, pp. 179–185.
Bierbrauer J. (2010), ‘New semifields, PN and APN functions’, Des. Codes Cryptogr. 54(3), 189–200.
Bierbrauer J. (2011), ‘Commutative semifields from projection mappings’, Des. Codes Cryptogr. 61(2), 187–196.
Biggs N. (1993), Algebraic Graph Theory, Cambridge Mathematical Library, second edn, Cambridge University Press, Cambridge.
Biggs N. L. & White A. T. (1979), Permutation Groups and Combinatorial Structures, Vol. 33 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge-New York.
Biliotti M., Jha V. & Johnson N. L. (2001), Foundations of Translation Planes, Vol. 243 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York.
Biliotti M. & Montinaro A. (2013), ‘On PGL(2, q)-invariant unitals embedded in Desarguesian or in Hughes planes’, Finite Fields Appl. 24, 66–87.
Blokhuis A. (1994), ‘On the size of a blocking set in PG(2,p)’, Combinatorica 14(1), 111–114.
Blokhuis A., Ball S., Brouwer A. E., Storme L. & Szőcnyi T. (1999), ‘On the number of slopes of the graph of a function defined on a finite field’, J. Combin. Theory Ser. A 86(1), 187–196.
Blokhuis A., Bruen A. A. & Thas J. A. (1988), ‘On M.D.S. codes, arcs in PG(n, q) with q even, and a solution of three fundamental problems of B. Segre’, Invent. Math. 92(3), 441–459.
Blokhuis A., Bruen A. & Thas J. A. (1990), ‘Arcs in PG(n, q), MDS-codes and three fundamental problems of B. Segre - some extensions’, Geom. Dedicata 35(1-3), 1–11.
Blokhuis A., De Boeck M., Mazzocca F. & Storme L. (2014), ‘The Kakeya problem: a gap in the spectrum and classification of the smallest examples’, Des. Codes Cryptogr. 72(1), 21–31.
Blokhuis A. & Mazzocca F. (2008), The finite field Kakeya problem, in Building bridges, Vol. 19 of Bolyai Soc. Math. Stud., Springer, Berlin, pp. 205–218.
Bollobás B. (1998), Modern Graph Theory, Vol. 184 of Graduate Texts in Mathematics, Springer-Verlag, New York.
Bollobás B. (2004), Extremal Graph Theory, Dover Publications., Inc., Mineola, NY.
Bombieri E. (1987), ‘Le grand crible dans la theorie analytique des nombres’, Asterisque (18), 103.
Bondy J. A. & Murty U. S. R. (2008), Graph Theory, Vol. 244 of Graduate Texts in Mathematics, Springer, New York.
Bondy J. A. & Simonovits M. (1974), ‘Cycles of even length in graphs’, J. Combinatorial Theory Ser. B 16, 97–105.
Bose R. C., Shrikhande S. S. & Parker E. T. (1960), ʿFurther results on the construction of mutually orthogonal Latin squares and the falsity of Euler's conjecture”, Canad. J. Math. 12, 189–203.
Brouwer A. E. (1985), ‘Some new two-weight codes and strongly regular graphs’, Discrete Appl. Math. 10(1), 111–114.
Brouwer A. E. & van Lint J. H. (1984), Strongly regular graphs and partial geometries, in Enumeration and Design (Waterloo, Ont., 1982), Academic Press, Toronto, ON, pp. 85–122.
Brown M. R. (2000a), ‘The determination of ovoids of PG(3, q) containing a pointed conic’, J. Geom. 67(1–2), 61–72.
Brown M. R. (2000b), ‘Ovoids of PG(3, q), q even, with a conic section’, J. London Math. Soc. (2) 62(2), 569–582.
Brown W. G. (1966), ‘On graphs that do not contain a Thomsen graph’, Canad. Math. Bull. 9, 281–285.
Bruck R. H. & Bose R. C. (1964), ‘The construction of translation planes from projective spaces’, J. Algebra 1, 85–102.
Bruck R. H. & Ryser H. J. (1949), ‘The nonexistence of certain finite projective plane’, Canad. J. Math. 1, 88–93.
Budaghyan L. & Helleseth T. (2011), ‘New commutative semifields defined by new PN multinomials’, Cryptogr. Commun. 3(1), 1–16.
Buekenhout F. (1976), ‘Existence of unitals in finite translation planes of order q2 with akerneloforder q’, Geometriae Dedicata 5(2), 189–194.
Buekenhout F., ed. (1995), Handbook of Incidence Geometry, North-Holland, Amsterdam.
Bush K. A. (1952), ‘Orthogonal arrays of index unity’, Ann. Math. Statistics 23, 426–434.
Calderbank R. & Kantor W. M. (1986), ‘The geometry of two-weight codes’, Bull. London Math. Soc. 18(2), 97–122.
Cameron P. J. (1994), Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, Cambridge.
Cameron P. J. (1999), Permutation Groups, Vol. 45 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge.
Cameron P. J. (2000), Projective and Polar Spaces, Vol.13 of QMW Maths Notes, Queen Mary and Westfield College School of Mathematical Sciences, London.
Cameron P. J. & van Lint J. H. (1975), Graph Theory, Coding Theory and Block Designs, London Mathematical Society Lecture Note Series, No. 19, Cambridge University Press, Cambridge-New York-Melbourne.
Cameron P. J. & van Lint J. H. (1991), Designs, Graphs, Codes and their Links, Vol. 22 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge.
Cardinali I. & De Bruyn B. (2013), ‘Spin-embeddings, two-intersection sets and two-weight codes’, Ars Combin. 109, 309–319.
Carlitz L. (1954), ‘Invariant theory of systems of equations in a finite field’, J. Analyse Math. 3, 382–413.
Casse R. (2006), Projective Geometry: An Introduction, Oxford University Press, Oxford.
Caullery F. & Schmidt K. -U. (2014), ‘On the classification of hyperovals’, arXiv:1403.2880v2.
Cherowitzo W. (1998), ‘α-flocks and hyperovals’, Geom. Dedicata 72(3), 221–246.
Cherowitzo W. E., O‘Keefe C. M. & Penttila T. (2003), 'A unified construction of finite geometries associated with q-clans in characteristic 2’, Adv. Geom. 3(1), 1–21.
Cherowitzo W., Penttila T., Pinneri I. & Royle G. F. (1996), ‘Flocks and ovals’, Geom. Dedicata 60(1), 17–37.
Cohen S. D. & Ganley M. J. (1982), ‘Commutative semifields, two-dimensional over their middle nuclei’, J. Algebra 75(2), 373–385.
Combarro E. F., Rua I. F. & Ranilla J. (2012), ‘Finite semifields with 74 elements’, Int. J. Comput. Math. 89(13–14), 1865–1878.
Cossidente A. (2010), ‘Embeddings of Un(q2) and symmetric strongly regular graphs’, J. Combin. Des. 18(4), 248–253.
Cossidente A., Durante N., Marino G., Penttila T. & Siciliano A. (2008), ‘The geometry of some two-character sets’, Des. Codes Cryptogr. 46(2), 231–241.
Cossidente A., Ebert G. L. & Korchmaros G. (2001), ‘Unitals in finite Desarguesian planes’, J. Algebraic Combin. 14(2), 119–125.
Cossidente A. & King O. H. (2010), ‘Some two-character sets’, Des. Codes Cryptogr. 56(2–3), 105–113.
Cossidente A. & Korchmáros G. (1998), ‘The algebraic envelope associated to a complete arc’, Rend. Circ. Mat. Palermo (2) Suppl. (51), 9–24.
Cossidente A. & Marino G. (2007), ‘Veronese embedding and two-character sets’, Des. Codes Cryptogr. 42(1), 103–107.
Cossidente A. & Penttila T. (2013), ‘Two-character sets arising fromgluings oforbits’, Graphs Combin. 29(3), 399–406.
Cossidente A. & Van Maldeghem H. (2007), ‘The simple exceptional group G2(q), q even, and two-character sets’, J. Combin. Theory Ser. A 114(5), 964–969.
Coxeter H. S. M. (1994), Projective Geometry, <Springer-Verlag, New York. Revised reprint of the second (1974) edition.
De Clerck F., De Winter S. & Maes T. (2012), ‘Partial flocks of the quadratic cone yielding Mathon maximal arcs’, Discrete Math. 312(16), 2421–2428.
De Clerck F. & Delanote M. (2000), ‘Two-weight codes, partial geometries and Steiner systems’, Des. Codes Cryptogr. 21(1–3), 87–98.
De Wispelaere A. & Van Maldeghem H. (2008), ‘Some new two-character sets in PG(5, q2) and a distance-2 ovoid in the generalized hexagon H(4)’, Discrete Math. 308(14), 2976–2983.
Delsarte P. (1972), ‘Weights of linear codes and strongly regular normed spaces’, Discrete Math. 3, 47–64.
Dembowski P. (1997), Finite Geometries, Classics in Mathematics, Springer-Verlag, Berlin. Reprint of the 1968 original.
Dembowski P. & Hughes D. R. (1965), ‘On finite inversive planes’, J. London Math. Soc. 40, 171–182.
Dempwolff U. (2008), ‘Semifield planes of order 81’, J. Geom. 89(1–2), 1–16.
Dempwolff U. & Müller P. (2013), ‘Permutation polynomials and translation planes of even order’, Adv. Geom. 13(2), 293–313.
Denniston R. H. F. (1969), ‘Some maximal arcs in finite projective planes’, J. Combin. Theory 6, 317–319.
Dickson L. E. (1896/97), ‘The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group’, Ann. ofMath. 11(1–6), 65–120.
Dickson L. E. (1906), ‘Linear algebras in which division is always uniquely possible’, Trans. Amer. Math. Soc. 7(3), 370–390.
Diestel R. (2005), Graph Theory, Vol. 173 of Graduate Texts in Mathematics, third edn, Springer-Verlag, Berlin.
Dirac G. A. (1952), ‘Some theorems on abstract graphs’, Proc. London Math. Soc. (3)2, 69–81.
Dixon J. D. & Mortimer B. (1996), Permutation Groups, Vol. 163 of Graduate Texts in Mathematics, Springer-Verlag, New York.
Donati G. & Durante N. (2012), ‘A group theoretic characterization of classical unitals’, J. Algebraic Combin. 36(1), 33–43.
Donati G., Durante N. & Siciliano A. (2014), ‘On unitals in PG(2, q2) stabilized by a homology group’, Des. Codes Cryptogr. 72(1), 135–139.
Dvir Z. (2009), ‘On the size of Kakeya sets in finite fields’, J. Amer. Math. Soc. 22(4), 1093–1097.
Dvir Z., Kopparty S., Saraf S. & Sudan M. (2013), ‘Extensions to the method of multiplicities, with applications to Kakeya sets and mergers’, SIAM J. Comput. 42(6), 2305–2328.
Ellenberg J. & Hablicsek M. (2014), ‘An incidence conjecture of Bourgain over fields of positive characteristic’, arXiv:1311.1479v1.
Erdős P. (1959), ‘Graph theory and probability’, Canad. J. Math. 11, 34–38.
Erdös P. & Stone A. H. (1946), ‘On the structure of linear graphs’, Bull. Amer. Math. Soc. 52, 1087–1091.
Feit W. & Higman G. (1964), ‘The nonexistence of certain generalized polygons’, J. Algebra 1, 114–131.
Frankl P. (1990), ‘Intersection theorems and mod p rank of inclusion matrices’, J. Combin. Theory Ser. A 54(1), 85–94.
Ftiredi Z. (1996a), ‘New asymptotics for bipartite Turan numbers’, J. Combin. Theory Ser. A 75(1), 141–144.
Füredi Z. (1996b), An upper bound on Zarankiewicz' problem, Combin. Probab. Comput. 5(1), 29–33.
Gács A. (2003), On a generalization of Rédei's theorem, Combinatorica 23(4), 585–598.
Glynn D.G. (1983), Two new sequences of ovals in finite Desarguesian planes of even order, in Combinatorial Mathematics, X (Adelaide, 1982), Vol. 1036 of Lecture Notes in Math., Springer, Berlin, pp. 217–229.
Glynn D.G. (1986), The nonclassical 10-arc of PG (4, 9), Discrete Math. 59(1-2), 43–51.
Glynn D.G. (1989), A condition for the existence of ovals in PG (2, q), q even, Geom. Dedicata 32(2), 247–252.
Godsil C. & Royle G. (2001), Algebraic Graph Theory, Vol. 207 of Graduate Texts in Mathematics, Springer-Verlag, New York.
Gould .R (2012), Graph Theory, Dover Publications, Inc., Mineola, NY.
Gow .R & Sheekey .J (2011), On primitive elements in finite semifields, Finite Fields Appl. 17(2), 194–204.
Graham R.L., Grötschel .M & Lovász .L, eds. (1995), Handbook of Combinatorics. Vol. 1, 2, Elsevier Science B.V., Amsterdam; MIT Press, Cambridge, MA.
Gupta .I, Narain .L & Veni Madhavan C.E. (2003), Cryptological applications of permutation polynomials, in Electronic Notes in Discrete Mathematics. Vol. 15, Vol. 15 of Electron. Notes Discrete Math., Elsevier, Amsterdam.p 93(electronic).
Guth .L & Katz N.H. (2010), Algebraic methods in discrete analogs of the Kakeya problem, Adv. Math. 225(5), 2828–2839.
Hamilton .N (2002a), Degree 8 maximal arcs in PG (2, 2h), h odd, J. Combin. Theory Ser. A 100(2), 265–276.
Hamilton .N (2002b), Strongly regular graphs from differences of quadrics, Discrete Math. 256(1-2), 465–469.
Hamilton .N & Mathon .R (2003), More maximal arcs in Desarguesian projective planes and their geometric structure, Adv. Geom. 3(3), 251–261.
Hamilton .N & Thas J.A. (2006), Maximal arcs in PG (2, q) and partial flocks of the quadratic cone, Adv. Geom. 6(1), 39–51.
Hill .R (1986), A First Course in Coding Theory, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York.
Hirschfeld J.W.P. (1971), Rational curves on quadrics over finite fields of characteristic two, Rend. Mat. (6) 4, 773–795 (1972).
Hirschfeld J.W.P. (1985), Finite Projective Spaces of Three Dimensions, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York.
Hirschfeld J.W. P. (1998), Projective Geometries Over Finite Fields, Oxford Mathematical Monographs, second edn, The Clarendon Press Oxford University Press, New York.
Hirschfeld J. W. P. & Korchmáros .G (1996), On the embedding of an arc into a conic in a finite plane, Finite Fields Appl. 2(3), 274–292.
Hirschfeld J.W. P. & Storme .L (2001), The packing problem in statistics, coding theory and finite projective spaces: update 2001, in Finite Geometries, Vol. 3 of Dev. Math., Kluwer Academic Publishers, Dordrecht, pp. 201–246.
Hirschfeld J.W. P. & Thas J.A. (1991), General Galois Geometries, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York.
Hughes D.R. & Piper F.C. (1973), Projective Planes, Graduate Texts inMathematics, Vol. 6, Springer-Verlag, New York-Berlin.
Hughes D.R. & Piper F.C. (1988), Design Theory, second edn, Cambridge University Press, Cambridge.
Huxley M.N. & Iwaniec .H (1975), Bombieri's theorem in short intervals, Mathematika 22(2), 188–194.
Johnson N.L., Jha .V & Biliotti M (2007), Handbook of Finite Translation Planes, Vol. 289 of Pure and Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL.
Jones G.A. & Jones J.M. (2000), Information and Coding Theory, Springer Undergraduate Mathematics Series, Springer-Verlag London Ltd., London.
Kantor W.M. (1979), Classical Groups From a Nonclassical Viewpoint, Oxford University, Mathematical Institute, Oxford.
Kantor W.M. (2006), Finite semifields, in Finite Geometries, Groups, and Computation, Walter de Gruyter GmbH & Co. KG, Berlin, pp. 103–114.
Knarr .N (1995), Translation Planes, Vol. 1611 of Lecture Notes in Mathematics, Springer-Verlag, Berlin. Foundations and construction principles.
Knuth D.E. (1965), Finite semifields and projective planes, J. Algebra 2, 182–217.
Kollár .J, Rónyai .L & Szabó T. (1996), Norm-graphs and bipartite Turán numbers, Combinatorica 16(3), 399–406.
Kövari .T, Sós V.T. & Turán .P (1954), On a problem of K. Zarankiewicz, Colloquium Math. 3, 50–57.
Laigle-Chapuy .Y (2007), Permutation polynomials and applications to coding theory, Finite Fields Appl. 13(1), 58–70.
Lam C.W.H., Thiel .L & Swiercz .S (1989), The nonexistence of finite projective planes of order 10, Canad. J. Math. 41(6), 1117–1123.
Lander E.S. (1983), Symmetric Designs: an Algebraic Approach, Vol. 74 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge.
Lang .S (1965), Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass.
Lavrauw .M & Sheekey .J (2013), Semifields from skew polynomial rings, Adv. Geom. 13(4), 583–604.
Lazebnik .F, Ustimenko V.A. & Woldar A.J. (1999), Polarities and 2k-cycle-free graphs, Discrete Math. 197/198, 503–513.
Levine .J & Chandler .R (1987), Some further cryptographic applications of permutation polynomials, Cryptologia 11(4), 211–218.
Lidl .R (1985), On cryptosystems based on polynomials and finite fields, in Advances in Cryptology (Paris, 1984), Vol. 209 of Lecture Notes in Comput. Sci., Springer, Berlin, pp. 10–15.
Lidl .R & Müller W.B. (1984a), A note on polynomials and functions in algebraic cryptography, Ars Combin. 17(A), 223–229.
Lidl .R & Müller W.B. (1984b), Permutation polynomials in RSA-cryptosystems, in Advances in Cryptology (Santa Barbara, Calif., 1983), Plenum, New York, pp. 293–301.
Lidl .R & Niederreiter .H (1997), Finite Fields, Vol. 20 of Encyclopedia of Mathematics and its Applications, second edn, Cambridge University Press, Cambridge.
Lindner C.C. & Rodger C.A. (2009), Design Theory, Discrete Mathematics and its Applications (Boca Raton), second edn, CRC Press, Boca Raton, FL.
Ling .S & Xing .C (2004), Coding Theory, A First Course, Cambridge University Press, Cambridge.
Lunardon .G (1999), Normal spreads, Geom. Dedicata 75(3), 245–261.
Lüneburg .H (1980), Translation Planes, Springer-Verlag, Berlin-New York.
Mac Lane .S & Birkhoff .G (1967), Algebra, The Macmillan Co., New York.
MacWilliams F.J. & Sloane N.J.A. (1977), The Theory of Error-Correcting Codes. I, North-Holland Publishing Co., Amsterdam.
Marino .G, Polverino .O & Trombetti .R (2007), On Fq-linear sets of PG (3, q3) and semifields, J. Combin. Theory Ser. A 114(5), 769–788.
Masuda A.M. & Zieve M.E. (2009), Permutation binomials over finite fields, Trans. Amer. Math. Soc. 361(8), 4169–4180.
Mathon .R (2002), New maximal arcs in Desarguesian planes, J. Combin. Theory Ser. A 97(2), 353–368.
McEliece R.J. (2004), The Theory of Information and Coding, Vol. 86 of Encyclopedia of Mathematics and its Applications, student edn, Cambridge University Press, Cambridge.
Metz .R (1979), On a class of unitals, Geom. Dedicata 8(1), 125–126.
Mullen G.L., ed. (2013), Handbook of Finite Fields, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL.
O'Keefe C.M. & Penttila .T (1992), Ovoids of PG (3, 16) are elliptic quadrics. II, J. Geom. 44(1-2), 140–159.
O'Keefe C.M., Penttila .T & Royle G.F. (1994), Classification of ovoids in PG (3, 32), J. Geom. 50(1-2), 143–150.
Ostrom T.G. (1970), Finite Translation Planes, Lecture Notes in Mathematics, Vol. 158, Springer-Verlag, Berlin-New York.
Oxley J.G. (1992), Matroid Theory, Oxford Science Publications, The Clarendon Press Oxford University Press, New York.
Parker E.T. (1959), Orthogonal latin squares, Proc. Nat. Acad. Sci. U.S.A. 45, 859–862.
Pauley .M & Bamberg .J (2008), A construction of one-dimensional affine flagtransitive linear spaces, Finite Fields Appl. 14(2), 537–548.
Payne S.E. (1985), A new infinite family of generalized quadrangles, in Proceedings of the Sixteenth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1985), Vol. 49, pp. 115–128.
Payne S.E., Penttila .T & Pinneri .I (1995), Isomorphisms between Subiaco q-clan geometries, Bull. Belg. Math. Soc. Simon Stevin 2(2), 197–222.
Payne S.E. & Thas J.A. (1984), Finite Generalized Quadrangles, Vol. 110 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA.
Penttila .T & Royle G.F. (1995), Sets of type (m, n) in the affine and projective planes of order nine, Des. Codes Cryptogr. 6(3), 229–245.
Penttila .T & Williams .B (2000), Ovoids of parabolic spaces, Geom. Dedicata 82(1-3), 1–19.
Polverino .O (2000), Small blocking sets in PG (2, p3), Des. Codes Cryptogr. 20(3), 319–324.
Rédei .L (1970), Lückenhafte Polynome über endlichen Körpern, Birkhäuser Verlag, Basel-Stuttgart.
Reed I.S. & Solomon .G (1960), Polynomial codes over certain finite fields, J. Soc. Indust. Appl. Math. 8, 300–304.
Roman .S (1997), Introduction to Coding and Information Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York.
Rúa I.F., Combarro E.F. & Ranilla .J (2009), Classification of semifields of order 64, J. Algebra 322(11), 4011–4029.
Rúa I.F., Combarro E.F. & Ranilla .J (2012), Determination of division algebras with 243 elements, Finite Fields Appl. 18(6), 1148–1155.
Schmidt .B (2002), Characters and Cyclotomic Fields in Finite Geometry, Vol. 1797 of Lecture Notes in Mathematics, Springer-Verlag, Berlin.
Schwartz J.T. (1980), Fast probabilistic algorithms for verification of polynomial identities, J. Assoc. Comput. Mach. 27(4), 701–717.
Segre .B (1955), Ovals in a finite projective plane, Canad. J. Math. 7, 414–416.
Segre .B (1957), Sui k-archi nei piani finiti di caratteristica due, Rev. Math. Pures Appl. 2, 289–300.
Segre .B (1962), Ovali e curve σ nei piani di Galois di caratteristica due., Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 32, 785–790.
Segre .B (1967), Introduction to Galois geometries, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 8, 133–236.
Semple J.G. & Kneebone G.T. (1998), Algebraic Projective Geometry, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York. Reprint of the 1979 edition.
Shafarevich I.R. (1994), Basic Algebraic Geometry. 1, second edn, Springer-Verlag, Berlin. Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid.
Sziklai .P (2008), On small blocking sets and their linearity, J. Combin. Theory Ser. A 115(7), 1167–1182.
Sziklai .P & Van de Voorde .G (2013), A small minimal blocking set in PG(n, pt), spanning a (t - 1)-space, is linear, Des. Codes Cryptogr. 68(1-3), 25–32.
Szőonyi .T (1997), Blocking sets in Desarguesian affine and projective planes, Finite Fields Appl. 3(3), 187–202.
Taylor D.E. (1992), The Geometry of the Classical Groups, Vol. 9 of Sigma Series in Pure Mathematics, Heldermann Verlag, Berlin.
Thas J.A. (1972), Ovoidal translation planes, Arch. Math. (Basel) 23, 110–112.
Thas J.A. (1974), Construction of maximal arcs and partial geometries, Geometriae Dedicata 3, 61–64.
Tits .J (1962), Ovoïdes et groupes de Suzuki, Arch. Math. 13, 187–198.
van Lint J.H. (1999), Introduction to Coding Theory, Vol. 86 of Graduate Texts in Mathematics, third edn, Springer-Verlag, Berlin.
van Lint J.H. & Wilson R.M. (2001), A Course in Combinatorics, second edn, Cambridge University Press, Cambridge.
Van Maldeghem .H (1998), Generalized Polygons, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel.
Voloch J.F. (1990), Arcs in projective planes over prime fields, J. Geom. 38(1-2), 198–200.
Voloch J.F. (1991), Complete arcs in Galois planes of nonsquare order, in Advances in Finite Geometries and Designs (Chelwood Gate, 1990), Oxford Science Publications, Oxford University Press, New York, pp. 401–406.
Voloshin V.I. (2009), Introduction to Graph Theory, Nova Science Publishers, Inc., New York.
Wan Z.X. (1993), Geometry of Classical Groups Over Finite Fields, Studentlitteratur, Lund; Chartwell-Bratt Ltd., Bromley.
Wan .Z-X. (2009), Design Theory, Higher Education Press, Beijing; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
Weil .A (1949), Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55, 497–508.
Weiner .Z (2004), On (k, pe)-arcs in Desarguesian planes, Finite Fields Appl. 10(3), 390–404.
Weyl .H (1997), The Classical Groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ. Their invariants and representations, fifteenth printing, Princeton Paperbacks.
Wielandt .H (1964), Finite Permutation Groups, Translated from the German by R. Bercov, Academic Press, New York–London.
Wilson R.A. (2009), The Finite Simple Groups, Vol. 251 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd, London.
Wilson R.M. (1990), A diagonal form for the incidence matrices of t-subsets vs. k-subsets, European J. Combin. 11(6), 609–615.
Xambó-Descamps .S (2003), Block Error-Correcting Codes, A Computational Primer, Universitext, Springer-Verlag, Berlin.
Zhou .Y & Pott .A (2013), A new family of semifields with 2 parameters, Adv. Math. 234, 43–60.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 547 *
Loading metrics...

Book summary page views

Total views: 806 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th January 2018. This data will be updated every 24 hours.