Skip to main content
SpringerLink
Log in
Book cover

Finite Geometries pp 201–246Cite as

The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001

  • Chapter

Part of the book series: Developments in Mathematics ((DEVM,volume 3))

Abstract

This article updates the authors’ 1998 survey [134] on the same theme that was written for the Bose Memorial Conference (Colorado, June 7–11, 1995). That article contained the principal results on the packing problem, up to 1995. Since then, considerable progress has been made on different kinds of subconfigurations.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L.M. Abatangelo, Calotte complete in uno spazio di Galois PG(3, q), q pari, Atti Sem. Mat. Fis. Univ. Modena 29 (1980), 215–221.

    MathSciNet  MATH  Google Scholar 

  2. V. Abatangelo, A class of complete ((q + 8)/3)-arcs in PG(2,2h) (h ≥ 6 even), Ars Combin. 16 (1983), 103–111.

    MathSciNet  MATH  Google Scholar 

  3. V. Abatangelo, Un nuovo procedimento per la costruzione di calotte di PG(3, q), q pari, Rend. Sem. Mat. Brescia 7 (1984), 19–25.

    MathSciNet  MATH  Google Scholar 

  4. V. Abatangelo, G. Korchmáros and B. Larato, Classification of maximal caps in PG(3, 5) different from elliptic quadrics, J. Geom. 57 (1996), 9–19.

    Article  MathSciNet  MATH  Google Scholar 

  5. A.H. Ali, Classification of arcs in the plane of order 13, Ph.D. Thesis, University of Sussex, 1993.

    Google Scholar 

  6. A.H. Ali, J.W.P. Hirschfeld and H. Kaneta, The automorphism group of a complete (q − 1)-arc in PG(2, q), J. Combin. Des. 2 (1994), 131–145.

    Article  MathSciNet  MATH  Google Scholar 

  7. A.H. Ali, J.W.P. Hirschfeld and H. Kaneta, On the size of arcs in projective spaces, IEEE Trans. Inform. Theory 41 (1995), 1649–1656.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Ball, On sets of points in finite planes, Ph.D. Thesis, University of Sussex, 1994.

    Google Scholar 

  9. S. Ball, On the size of a triple blocking set in PG(2, q),European J. Combin. 17 (1996), 427–435.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Ball, Multiple blocking sets and arcs in finite planes, J. London Math. Soc. 54 (1996), 581–593.

    MathSciNet  MATH  Google Scholar 

  11. S. Ball, On small complete arcs in a finite plane, Discrete Math. 174 (1997), 29–34.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Ball, Some notes on arcs and multi-arcs in projective and affine planes, unpublished manuscript (2000).

    Google Scholar 

  13. S. Ball and A. Blokhuis, On the size of a double blocking set in PG(2, q), Finite Fields Appl. 2 (1996), 125–137.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Ball and A. Blokhuis, An easier proof of the maximal arcs conjecture, Proc. Amer. Math. Soc. 126 (1998), 3377–3380.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Ball and A. Blokhuis, On the incompleteness of (k, n)-arcs in Desarguesian planes of order q where n divides q, Geom. Dedicata 74 (1999), 325–332.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Ball, A. Blokhuis and F. Mazzocca, Maximal arcs in Desarguesian planes of odd order do not exist, Combinatorica 17 (1997), 31–47.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Ball, R. Hill, I.N. Landjev and H.N. Ward, On (q 2 + q + 2, q + 2)-arcs in the projective plane PG(2, q), in preparation.

    Google Scholar 

  18. J. Barát, A. Del Fra, S. Innamorati and L. Storme, Minimal blocking sets in PG(2, 8) and maximal partial spreads in PG(3, 8), in preparation.

    Google Scholar 

  19. J. Barát, Y. Edel, R. Hill, C. Jones, I.N. Landjev and L. Storme, Caps in PG(5, 3) and PG(6, 3), in preparation.

    Google Scholar 

  20. J. Barát and L. Storme, Multiple blocking sets in PG(n, q),n ≥ 3, submitted.

    Google Scholar 

  21. A. Barlotti, Un’ estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital. 10 (1955), 96–98.

    MathSciNet  MATH  Google Scholar 

  22. A. Barlotti, Su {k; n}-archi di un piano lineare finito, Boll. Un. Mat. Ital. 11 (1956), 553–556.

    MathSciNet  MATH  Google Scholar 

  23. A. Barlotti, Some topics in finite geometrical structures, Institute of Statistics Mimeo Series 439, Univ. of North Carolina, 1965.

    Google Scholar 

  24. A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata 9 (1980), 130–157.

    Article  MathSciNet  Google Scholar 

  25. J. Bierbrauer, (k, n)-arcs of maximal size in the plane of order 8, Unpublished manuscript (1988).

    Google Scholar 

  26. J. Bierbrauer and Y. Edel, A family of caps in projective 4-space in odd characteristic, Finite Fields Appl. 6 (2000), 283–293.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Bierbrauer and Y. Edel, Bounds on affine caps, J. Combin. Des., to appear.

    Google Scholar 

  28. A. Blokhuis, Note on the size of a blocking set in PG(2, p), Combinatorica 14 (1994), 111–114.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Blokhuis, On multiple nuclei and a conjecture of Lunelli and Sce, Bull. Belg. Math. Soc. 3 (1994), 349–353.

    Google Scholar 

  30. A. Blokhuis, Blocking sets in Desarguesian planes, Paul Erdős is Eighty, Volume 2 (eds. D. Miklós, V.T. Sós and T. Szőnyi), Bolyai Soc. Math. Studies 2 (1996), 133–155.

    Google Scholar 

  31. A. Blokhuis, S. Sall, A.E. Brouwer, L. Storme and T. Szőnyi, On the number of slopes of the graph of a function defined on a finite field, J. Combin. Theory Ser. A 86 (1999), 187–196.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Blokhuis and K. Metsch, Large minimal blocking sets, strong representative systems, and partial unitals, Finite Geometry and Combinatorics (eds. F. De Clerck et al.), London Math. Soc. Lecture Note Series 191, Cambridge Univer-sity Press, Cambridge, 1993, 37–52.

    Chapter  Google Scholar 

  33. A. Blokhuis, R. Pellilcaan and T. Szőnyi, Blocking sets of almost Rédei type, J. Combin. Theory Ser. A 78 (1997), 141–150.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Blokhuis, L. Storme and T. Szőnyi, Lacunary polynomials, multiple blocking sets and Baer subplanes, J. London Math. Soc. 60 (1999), 321–332.

    Article  MathSciNet  Google Scholar 

  35. M. Bokler, Minimal blocking sets in projective spaces of square order, Des. Codes Cryptogr., to appear.

    Google Scholar 

  36. M. Bokler and K. Metsch, On the smallest minimal blocking sets in projective spaces generating the whole space, Contrib. Algebra Geom., to appear.

    Google Scholar 

  37. E. Boros and T. Szőnyi, On the sharpness of a theorem of B. Segre, Combinatorica 6 (1986), 261–268.

    Article  MathSciNet  MATH  Google Scholar 

  38. R.C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā 8 (1947), 107–166.

    MATH  Google Scholar 

  39. R.C. Bose, On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements, Golden Jubilee Commemoration Volume (1958–1959), Calcutta Math. Soc. (1959), 341–354.

    Google Scholar 

  40. R.C. Bose, On some connections between the design of experiments and infor-mation theory, Bull. Inst. Internat. Statist. 38 (1961), 257–271.

    MathSciNet  MATH  Google Scholar 

  41. R.C. Bose and R.C. Burton, A characterization of flat spaces in a finite projective geometry and the uniqueness of the Hamming and the MacDonald codes, J. Combin. Theory 1 (1966), 96–104.

    Article  MathSciNet  MATH  Google Scholar 

  42. R.C. Bose and J.N. Srivastava, On a bound useful in the theory of factorial designs and error correcting codes, Ann. Math. Statist. 35 (1964), 408–414.

    Article  MathSciNet  MATH  Google Scholar 

  43. A.A. Bruen, Baer subplanes and blocking sets, Bull. Amer. Math. Soc. 76 (1970), 342–344.

    Article  MathSciNet  MATH  Google Scholar 

  44. A.A. Bruen, Blocking sets in finite projective planes, SIAM J. Appl. Math. 21 (1971), 380–392.

    Article  MathSciNet  MATH  Google Scholar 

  45. A.A. Bruen, Polynomial multiplicities over finite fields and intersection sets, J. Combin. Theory Ser. A 60 (1992), 19–33.

    Article  MathSciNet  MATH  Google Scholar 

  46. A.A. Bruen and J.A. Thas, Blocking sets, Geom. Dedicata 6 (1977), 193–203.

    MathSciNet  MATH  Google Scholar 

  47. K.A. Bush, Orthogonal arrays of index unity, Ann. Math. Statist. 23 (1952), 426–434.

    Article  MathSciNet  MATH  Google Scholar 

  48. A.R. Calderbank and P.C. Fishburn, Maximal three-independent subsets of {0, 1, 2}n, Des. Codes Cryptogr. 4 (1994), 203–211.

    Article  MathSciNet  MATH  Google Scholar 

  49. L.R.A. Casse, A solution to Beniamino Segre’s ‘Problem I r,q ’ for q even, Atti Accad. Naz. Lincei Rend. 46 (1969), 13–20.

    MathSciNet  MATH  Google Scholar 

  50. L.R.A. Casse and D.G. Glynn, The solution to Beniamino Segre’s problem I r,q , r = 3, q = 2h, Geom. Dedicata 13 (1982), 157–164.

    Article  MathSciNet  MATH  Google Scholar 

  51. L.R.A. Casse and D.G. Glynn, On the uniqueness of (q+1)4-arcs of PG(4, q), q = 2h, h ≥ 3, Discrete Math. 48 (1984), 173–186.

    Article  MathSciNet  MATH  Google Scholar 

  52. J.M. Chao, On the size of a cap in PG(n, q) with q even and n ≥ 3, Geom. Dedicata 74 (1999), 91–94.

    Article  MathSciNet  MATH  Google Scholar 

  53. J.M. Chao, On the maximum size of a \((k,\sqrt {q} + 1)\)-arc in PG(2, q), with q square, Atti Sem. Mat. Fis. Univ. Modena 47 (1999), 529–531.

    MathSciNet  MATH  Google Scholar 

  54. J.M. Chao and H. Kaneta, A complete 24-arc in PG(2, 29) with automorphism group PSL(2, 7), Rend. Mat. Appl. 16 (1996), 537–544.

    MathSciNet  MATH  Google Scholar 

  55. J.M. Chao and H. Kaneta, Classical arcs in PG(r, q) for 11 ≤ q ≤ 19, Discrete Math. 174 (1997), 87–94.

    Article  MathSciNet  MATH  Google Scholar 

  56. J.M. Chao and H. Kaneta, Classical arcs in PG(r, q) for 23 ≤ q ≤ 27, submitted.

    Google Scholar 

  57. W.E. Cherowitzo, Hyperovals in Desarguesian planes of even order, Ann. Discrete Math. 37 (1988), 87–94.

    Article  MathSciNet  Google Scholar 

  58. W.E. Cherowitzo, α-flocks and hyperovals, Geom. Dedicata 72 (1998), 221–246.

    Article  MathSciNet  MATH  Google Scholar 

  59. W.E. Cherowitzo, C.M. O’Keefe and T. Penttila, A unified construction of finite geometries in characteristic two, submitted.

    Google Scholar 

  60. W.E. Cherowitzo, T. Penttila, I. Pinneri and G.F. Royle, Flocks and ovals, Geom. Dedicata 60 (1996), 17–37.

    Article  MathSciNet  MATH  Google Scholar 

  61. Z. Ciechanowicz, Ph.D. Thesis, University of London, 1980.

    Google Scholar 

  62. G.D. Cohen, M.G. Karpovsky, H.F. Mattson, and J.R. Schatz, Covering radius: survey and recent results, IEEE Trans. Inform. Theory 31 (1985), 328–343.

    Article  MathSciNet  MATH  Google Scholar 

  63. G.D. Cohen, S.N. Litsyn, H. Honkala and H.F. Mattson, Weighted coverings and packings, IEEE Trans. Inform. Theory 41 (1995), 1856–1867.

    Article  MathSciNet  MATH  Google Scholar 

  64. A. Cossu, Su alcune proprietà dei {k, n}-archi di un piano proiettivo sopra un corpo finito, Rend. Mat. e Appl. 20 (1961), 271–277.

    MathSciNet  MATH  Google Scholar 

  65. A.A. Davydov and P.R.J. Östergard, Recursive constructions of complete caps, J. Statist. Planning Infer., to appear.

    Google Scholar 

  66. A.A. Davydov and L.M. Tombak, Quasiperfect linear binary codes with distance 4 and complete caps in projective geometry, Problems Inform. Transmission 25 (1990), 265–275.

    MathSciNet  Google Scholar 

  67. M.A. de Boer, Almost MDS codes, Des. Codes Cryptogr. 9 (1996), 143–155.

    MathSciNet  MATH  Google Scholar 

  68. A. Del Fra and S. Innamorati, private communication (2000).

    Google Scholar 

  69. R.H.F. Denniston, Some maximal arcs in finite projective planes, J. Combin. Theory 6 (1969), 317–319.

    Article  MathSciNet  MATH  Google Scholar 

  70. S.M. Dodunekov and I.N. Landjev, On near-MDS codes, J. Geom. 54 (1995), 30–43.

    Article  MathSciNet  MATH  Google Scholar 

  71. Y. Edel, Private communication.

    Google Scholar 

  72. Y. Edel and J. Bierbrauer, 41 is the largest size of a cap in PG(4, 4), Des. Codes Cryptogr. 16 (1999), 51–60.

    Article  MathSciNet  Google Scholar 

  73. Y. Edel and J. Bierbrauer, Recursive constructions for large caps, Bull. Belg. Math. Soc. Simon Stevin 6 (1999), 249–258.

    MathSciNet  MATH  Google Scholar 

  74. Y. Edel and J. Bierbrauer, A family of caps in projective 4-space in characteristic 2, Congr. Numer. 141 (1999), 191–202.

    MathSciNet  MATH  Google Scholar 

  75. Y. Edel and J. Bierbrauer, Large caps in small spaces, Des. Codes Cryptogr., to appear.

    Google Scholar 

  76. Y. Edel, L. Storme and P. Sziklai, New upper bounds for the sizes of caps in PG(N, 5) and PG(N, 7), in preparation.

    Google Scholar 

  77. G. Faina, Complete caps having less than (q 2 + 1)/2 points in common with an elliptic quadric of PG(3, q), q odd, Rend. Mat. 8 (1988), 277–281.

    MathSciNet  MATH  Google Scholar 

  78. G. Faina, S. Marcugini, A. Milani and F. Pambianco, The spectrum of the values k for which there exists a complete k-arc in PG(2, q) for q ≤ 23, Ars Combin. 47 (1997), 3–11.

    MathSciNet  MATH  Google Scholar 

  79. G. Faina, S. Marcugini, A. Milani and F. Pambianco, The sizes k of the complete k-caps in PG(n,q), for small q and 3 ≤ n ≤ 5, Ars Combin 50 (1998), 225–243.

    MathSciNet  Google Scholar 

  80. G. Faina and F. Pambianco, Fasci di coniche e archi completi, preprint (1995).

    Google Scholar 

  81. G. Faina and F. Parnbianco, A class of complete k-caps in PG(3, q) for q an odd prime, J. Geom. 57 (1996), 93–105.

    Article  MathSciNet  MATH  Google Scholar 

  82. G. Faina and F. Pambianco, Small complete k-caps in PG(r, q), r ≥ 3, Discrete Math. 174 (1997), 114–123.

    Article  MathSciNet  Google Scholar 

  83. G. Faina and F Pambianco, On some 10-arcs for deriving the minimum order for complete arcs in small projective planes, Discrete Math. 208/209 (1999), 261–271.

    Article  MathSciNet  Google Scholar 

  84. P.G. Farrell, Linear binazy anticodes, Electronics Letters 6 (1970), 419–421.

    Article  MATH  Google Scholar 

  85. G. Fellegara, Gli ovaloidi di uno spazio tridimensionale di Galois di ordine 8, Atti Accad. Naz. Lincei Rend. 32 (1962), 170–176.

    MathSciNet  MATH  Google Scholar 

  86. S. Ferret and L. Storme, unpublished manuscript (2001).

    Google Scholar 

  87. S. Ferret and L. Storme, Minihypers and linear codes meeting the Griesmer bound: improvements to results of Ramada, Helleseth and Maekawa, submitted.

    Google Scholar 

  88. J.C. Fisher, J.W.P. Hirschfeld and J.A. Thas, Complete arcs in planes of square order, Ann. Discrete Math. 30 (1986), 243–250.

    MathSciNet  Google Scholar 

  89. R.A. Fisher, The theory of confounding in factorial experiments in relation to the theory of groups, Ann. Eugen. London 11 (1942), 341–353.

    Article  Google Scholar 

  90. R.A. Fisher, A system of confounding for factors with more than two alternatives giving completely orthogonal cubes and higher powers, Ann. Eugen. London 12 (1945), 283–290.

    Article  Google Scholar 

  91. E.M. Gabidulin, A.A. Davydov and L.M. Tombak, Linear codes with covering radius 2 and other new covering codes, IEEE Trans. Inform. Theory 37 (1991), 219–224.

    Article  MathSciNet  MATH  Google Scholar 

  92. A. Gács, A remark on blocking sets of almost Rédei type, J. Geom. 60 (1997), 65–73.

    Article  MathSciNet  MATH  Google Scholar 

  93. A. Gács, On a generalization of Rédei’s theorem, submitted.

    Google Scholar 

  94. A. Gács, The Rédei method applied to finite geometry, Ph.D. Thesis, Eötvös Loránd University, Budapest, Hungary, 1997.

    Google Scholar 

  95. A. Gács, P. Sziklai and T. Szőnyi, Two remarks on blocking sets and nuclei in planes of prime order, Des. Codes Cryptogr. 10 (1997), 29–39.

    Article  MathSciNet  MATH  Google Scholar 

  96. A. Gács and T. Szőnyi, Double blocking sets and Baer-subplanes of odd square order, unpublished manuscript (1995).

    Google Scholar 

  97. M. Giulietti, Small complete caps in PG(2, q), for q an odd square, J. Geom. 69 (2000), 110–116.

    Article  MathSciNet  MATH  Google Scholar 

  98. M. Giulietti, On plane arcs contained in cubic curves, submitted.

    Google Scholar 

  99. D.G. Glynn, Two new sequences of ovals in finite Desarguesian planes of even order, Combinatorial Mathematics X (ed. L.R.A. Casse), Lecture Notes in Mathematics 1036, Springer, 1983, 217–229.

    Chapter  Google Scholar 

  100. D.G. Glynn, The non-classical 10-arc of PG(4, 9), Discrete Math. 59 (1986), 43–51.

    Article  MathSciNet  MATH  Google Scholar 

  101. C.E. Gordon, Orbits of arcs in projective spaces, Finite Geometry and Cornbinatorics, London Math. Soc. Lecture Note Series 191, Cambridge University Press, Cambridge, 1993, 161–174.

    Google Scholar 

  102. P. Govaerts and L. Storme, On a particular class of minihypers of PG(N, q) and its applications. I: The result for general q, submitted.

    Google Scholar 

  103. P. Govaerts and L. Storme, On a particular class of minihypers of PG(N, q) and its applications. II: Improvements for q square, submitted.

    Google Scholar 

  104. J.H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop. 4 (1960), 532–542.

    Article  MathSciNet  MATH  Google Scholar 

  105. P. Gronchi, I blocking sets ed il packing problem, Boll. Un. Mat. Ital. 7-A (1993), 227–236.

    MathSciNet  Google Scholar 

  106. B.R. Gulati and E.G. Kounias, On bounds useful in the theory of symmetrical factorial designs, J. Roy. Statist. Soc. Ser. B 32 (1970), 123–133.

    MathSciNet  Google Scholar 

  107. É. Hadnagy, Small complete arcs in PG(2, p), Finite Fields Appl. 5 (1999), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  108. É. Hadnagy and T. Szőnyi, On embedding partial unitals and large (k, n)-arcs, Ars Combin., to appear.

    Google Scholar 

  109. H.-R. Halder and W. Heise, On the existence of finite chain-m-structures and k-arcs in finite projective space, Geom. Dedicata 3 (1974), 483–486.

    MathSciNet  Google Scholar 

  110. M. Hall, Ovals in the Desarguesian plane of order 16, Ann. Mat. Pura Appl. 102 (1975), 159–176.

    Article  MathSciNet  MATH  Google Scholar 

  111. N. Hamada, Characterization resp. nonexistence of certain q-ary linear codes attaining the Griesmer bound, Bull. Osaka Women’s Univ. 22 (1985), 1–47.

    MathSciNet  Google Scholar 

  112. N. Hamada, Characterization of minihypers in a finite projective geometry and its applications to error-correcting codes, Bull. Osaka Women’s Univ. 24 (1987), 1–24.

    Google Scholar 

  113. N. Hamada, A characterization of some [n, k, d; q]-codes meeting the Griesmer bound using a minihyper in a finite projective geometry, Discrete Math. 116 (1993), 229–268.

    Article  MathSciNet  MATH  Google Scholar 

  114. N. Hamada, The nonexistence of some quaternary linear codes meeting the Griesmer bound and the bounds for n 4(5, d), 1 ≤ d ≤ 256, Math. Japonica 43 (1996), 7–21.

    MathSciNet  MATH  Google Scholar 

  115. N. Hamada, The nonexistence of some ternary linear codes of dimension 6 and the bounds for n 3(6, d), 1 ≤ d ≤ 243, Math. Japonica 43 (1996), 577–593.

    MathSciNet  MATH  Google Scholar 

  116. N. Hamada and T. Helleseth, A characterization of some q-ary codes (q > (h − 1)2,h ≥ 3) meeting the Griesmer bound, Math. Japonica 38 (1993), 925–940.

    MathSciNet  MATH  Google Scholar 

  117. N. Hamada and T. Maekawa, A characterization of some q-ary codes (q > (h − 1)2, h ≥ 3) meeting the Griesmer bound: Part 2, Math. Japonica 46 (1997), 241–252.

    MathSciNet  MATH  Google Scholar 

  118. R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.

    MATH  Google Scholar 

  119. U. Heim, On t-blocking sets in projective spaces, unpublished manuscript (1994).

    Google Scholar 

  120. R. Hill, On the largest size of cap in S 5,3, Atti Accad. Naz. Lincei Rend. 54 (1973), 378–384.

    Google Scholar 

  121. R. Hill, Caps and codes, Discrete Math. 22 (1978), 111–137.

    Article  MathSciNet  MATH  Google Scholar 

  122. R. Hill, On Pellegrino’s 20-caps in S 4,3, Combinatorial Geometries and their Applications (Rome 1981), Ann. Discrete Math. 18 (1983), 443–448.

    Google Scholar 

  123. R. Hill, Some problems concerning (k, n)-arcs in finite projective planes, Rend. Sem. Mat. Brescia 7 (1984), 367–383.

    MATH  Google Scholar 

  124. R. Hill and J.R.M. Mason, On (k, n)-arcs and the falsity of the Lunelli-Sce conjecture, Finite Geometries and Designs, London Math. Soc. Lecture Note Series 49, Cambridge University Press, Cambridge, 1981, 153–168.

    Google Scholar 

  125. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, Oxford, 1979.

    MATH  Google Scholar 

  126. J.W.P. Hirschfeld, Caps in elliptic quadrics, Ann. Discrete Math. 18 (1983), 449–466.

    MathSciNet  MATH  Google Scholar 

  127. J.W.P. Hirschfeld, Maximum sets in finite projective spaces, Surveys in Combinatorics, London Math. Soc. Lecture Note Series 82, Cambridge University Press, Cambridge, 1983, 55–76.

    Google Scholar 

  128. J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985.

    MATH  Google Scholar 

  129. J.W.P. Hirschfeld, Complete arcs, Discrete Math. 174 (1997), 177–184.

    Article  MathSciNet  MATH  Google Scholar 

  130. J.W.P. Hirschfeld, Projective Geometries over Finite Fields (Second Edition), Oxford University Press, Oxford, 1998.

    MATH  Google Scholar 

  131. J.W.P. Hirschfeld and G. Korchmáros, On the embedding of an arc into a conic in a finite plane, Finite Fields Appl. 2 (1996), 274–292.

    Article  MathSciNet  MATH  Google Scholar 

  132. J.W.P. Hirschfeld and G. Korchmáros, On the number of rational points on an algebraic curve over a finite field, Bull. Belg. Math. Soc. Simon Stevin 5 (1998), 313–340.

    MathSciNet  MATH  Google Scholar 

  133. J.W.P. Hirschfeld and G. Korchmáros, Arcs and curves over a finite field, Finite Fields Appl. 5 (1999), 393–408.

    MATH  Google Scholar 

  134. J.W.P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces, J. Statist. Planning Infer. 72 (1998), 355–380.

    Article  MathSciNet  MATH  Google Scholar 

  135. J.W.P. Hirschfeld and T. Szőnyi, Constructions of large arcs and blocking sets in finite planes, European J. Combin. 12 (1991), 499–511.

    MathSciNet  MATH  Google Scholar 

  136. J.W.P. Hirschfeld and J.A. Thas, Linear independence in finite spaces, Geom. Dedicata 23 (1987), 15–31.

    Article  MathSciNet  MATH  Google Scholar 

  137. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, 1991.

    MATH  Google Scholar 

  138. S. Innamorati, The non-existence of certain large minimal blocking sets, Mitt. Math. Sem. Giessen 235 (1998), 1–23.

    MathSciNet  MATH  Google Scholar 

  139. S. Innamorati and A. Maturo, On irreducible blocking sets in projective planes, Ratio Math. 2 (1991), 151–155.

    Google Scholar 

  140. H. Kaneta, private communication (1995).

    Google Scholar 

  141. H. Kaneta, private communication (1998).

    Google Scholar 

  142. H. Kaneta and T. Maruta, An elementary proof and an extension of Thas’ theorem on k-arcs, Math. Proc. Cambridge Philos. Soc. 105 (1989), 459–462.

    MathSciNet  MATH  Google Scholar 

  143. B.C. Kestenband, A family of complete arcs in finite projective planes, Colloq. Math. 57 (1989), 59–67.

    MathSciNet  MATH  Google Scholar 

  144. J.H. Kim and V.H. Vu, Small complete arcs in projective planes, submitted.

    Google Scholar 

  145. G. Korchmáros, New examples of complete k-arcs in PG(2, q), European J. Combin. 4 (1983), 329–334.

    MathSciNet  MATH  Google Scholar 

  146. P. Kustaanheimo, A note on a finite approximation of the Euclidean plane geometry, Soc. Sci. Fenn. Comment. Phys.-Math. 15, No. 19 (1950).

    MathSciNet  Google Scholar 

  147. P. Lisonek, Computer-assisted studies in algebraic combinatorics, Ph.D. Thesis, Research Institute for Symbolic Computation, J. Kepler University Linz, 1994.

    Google Scholar 

  148. L. Lombardo-Radice, Sul prod ema dei k-archi completi di S 2,q , Boll. Un. Mat. Ital. 11 (1956), 178–181.

    MathSciNet  MATH  Google Scholar 

  149. L. Lovász and A. Schrijver, Remarks on a theorem of Rédei, Studia Scient. Math. Hungar. 16 (1981), 449–454.

    MATH  Google Scholar 

  150. L. Lovász and T. Szőnyi, Multiple blocking sets and algebraic curves, abstract from Finite Geometry and Combinatorics (Third International Conference at Deinze (Belgium), May 18–24, 1997).

    Google Scholar 

  151. G. Lunardon, Normal spreads, Geom. Dedicata 75 (1999), 245–261.

    Article  MathSciNet  MATH  Google Scholar 

  152. L. Lunelli and M. Sce, Considerazione aritmetiche e risultati sperimentali sui {K; n} q -archi, Ist. Lombardo Accad. Sci. Rend. A 98 (1964), 3–52.

    MathSciNet  MATH  Google Scholar 

  153. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.

    MATH  Google Scholar 

  154. S. Marcugini, A. Milani and F. Pambianco, Maximal (n, 3)-arcs in PG(2, 11), Discrete Math. 208/209 (1999), 421–426.

    Article  MathSciNet  Google Scholar 

  155. S. Marcugini, A. Milani and F. Pambianco, Maximal (n, 3)-arcs in PG(2, 13), in preparation.

    Google Scholar 

  156. S. Marcugini, A. Milani and F. PaMbianco, Complete arcs in PG(2, 25): the spectrum of the sizes and the classification of the smallest complete arcs, in preparation.

    Google Scholar 

  157. S. Marcugini, A. Milani and F. Pambianco, Minimal complete arcs in PG(2, q), q ≤ 29, in preparation.

    Google Scholar 

  158. T. Maruta and H. Kaneta, On the uniqueness of (q + 1)-arcs of PG(5, q), q = 2h, h ≥ 4, Math. Proc. Cambridge Philos. Soc. 110 (1991), 91–94.

    Article  MathSciNet  MATH  Google Scholar 

  159. J.R.M. Mason, On the maximum sizes of certain (k, n)-arcs in finite projective geometries, Math. Proc. Cambridge Philos. Soc. 91 (1982), 153–170.

    Article  MathSciNet  MATH  Google Scholar 

  160. J.R.M. Mason, A class of ((p n − p m)(p n − 1), p n − p m)-arcs in PG(2, p n), Geom. Dedicata 15 (1984), 355–361.

    Article  MathSciNet  MATH  Google Scholar 

  161. R. Meshulam, On subsets of finite abelian groups with no 3-term arithmetic progression, J. Combin. Theory Ser. A 71 (1995), 168–172.

    Article  MathSciNet  MATH  Google Scholar 

  162. A. Milani, S. Marcugini and F. Pambianco, Computer aided determination of very small axcs in PG(2, q), q ≤ 67, preprint.

    Google Scholar 

  163. A.C. Mukhopadhyay, Lower bounds on m t (r, s), J. Combin. Theory Ser. A 25 (1978), 1–13.

    Article  MATH  Google Scholar 

  164. G.P. Nagy and T. Szőnyi, Caps in finite projective spaces of odd order, J. Geom. 59 (1997), 103–113.

    Article  MathSciNet  MATH  Google Scholar 

  165. C.M. O’Keefe and T. Penttila, Ovoids of PG(3, 16) are elliptic quadrics, J. Geom. 38 (1990), 95–106.

    Article  MathSciNet  MATH  Google Scholar 

  166. C.M. O’Keefe and T. Penttila, Hyperovals in PG(2, 16), European J. Combin. 12 (1991), 51–59.

    MathSciNet  MATH  Google Scholar 

  167. C.M. O’Keefe and T. Penttila, A new hyperoval in PG(2, 32), J. Geom. 44 (1992), 117–139.

    Article  MathSciNet  MATH  Google Scholar 

  168. C.M. O’Keefe and T. Penttila, Ovoids of PG(3, 16) axe elliptic quadrics, II, J. Geom. 44 (1992), 140–159.

    Article  MathSciNet  MATH  Google Scholar 

  169. C.M. O’Keefe, T. Penttila and G.F. Royle, Classification of ovoids in PG(3, 32), J. Geom. 50 (1994), 143–150.

    Article  MathSciNet  MATH  Google Scholar 

  170. C.M. O’Keefe and L. Storme, Arcs fixed by A 5 and A 6, J. Geom. 55 (1996), 123–138.

    Article  MathSciNet  MATH  Google Scholar 

  171. P.R.J. Östergard, Computer search for small complete caps, J. Geom. 69 (2000), 172–179.

    Article  MathSciNet  MATH  Google Scholar 

  172. S. Packer, Sets of odd type and caps in Galois geometries of order four, D.Phil. Thesis, University of Sussex, 1995.

    Google Scholar 

  173. F. Pambianco and L. Storme, Small complete caps in spaces of even characteristic, J. Coubbin. Theory Ser. A 75 (1996), 70–84.

    Article  MathSciNet  MATH  Google Scholar 

  174. F Pambianco and L. Storme, unpublished manuscript (1995).

    Google Scholar 

  175. G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. Un. Mat. Ital. 10 (1955), 507–513.

    MathSciNet  MATH  Google Scholar 

  176. S.E. Payne, A new infinite family of generalized quadrangles, Congr. Numer. 49 (1985), 115–128.

    MathSciNet  Google Scholar 

  177. S.E. Payne, A tensor product action on q-clan generalized quadrangles with q = 2e, Linear Algebra Appl. 226–228 (1995), 115–137.

    Article  Google Scholar 

  178. S.E. Payne, T. Penttila and I. Pinneri, Isomorphisms between Subiaco q-clan geometries, Bull. Belg. Math. Soc. 2 (1995), 197–222.

    MathSciNet  MATH  Google Scholar 

  179. S.E. Payne, T. Penttila and G.F. Royle, Constructing a cyclic q-clan, Mostly Finite Geometries (ed. N.L. Johnson), Marcel Dekker, New York, 1997, 365–378.

    Google Scholar 

  180. G. Pellegrino, Sul massimo ordine delle calotte in S 4,3, Matematiche (Catania) 25 (1970), 1–9.

    MathSciNet  Google Scholar 

  181. G. Pellegrino, Sulle calotte complete, non ovaloidi, dello spazio PG(3, q), q dispari, Rend. Circ. Mat. Palermo 47 (1998), 141–168.

    Article  MathSciNet  MATH  Google Scholar 

  182. T. Penttila, private communication (2000).

    Google Scholar 

  183. T. Penttila and C.E. Praeger, Ovoids and translation ovals, Proc. London Math. Soc. 56 (1997), 607–624.

    Article  MathSciNet  MATH  Google Scholar 

  184. T. Penttila and G.F. Royle, Classification of hyperovals in PG(2, 32), J. Geom. 50 (1994), 151–158.

    Article  MathSciNet  MATH  Google Scholar 

  185. T. Penttila and G.F. Royle, On hyperovals in small projective planes, J. Geom. 54 (1995), 91–104.

    Article  MathSciNet  MATH  Google Scholar 

  186. T. Penttila and G.F. Royle, private communication (1995).

    Google Scholar 

  187. P. Polito and O. Polverino, On small blocking sets, Combinatorica 18 (1998), 133–137.

    Article  MathSciNet  MATH  Google Scholar 

  188. O. Polverino, Small blocking sets and complete k-arcs in PG(2, p 3), Discrete Math. 208/209 (1999), 469–476.

    Article  MathSciNet  Google Scholar 

  189. O. Polverino, Small blocking sets in PG(2, p 3), Des. Codes Cryptogr. 20 (2000), 319–324.

    Article  MathSciNet  MATH  Google Scholar 

  190. O. Polverino and L. Storme, Small blocking sets in PG(2, q 3), submitted.

    Google Scholar 

  191. B. Qvist, Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn. Ser. A 134 (1952).

    Google Scholar 

  192. A.R. Sadeh, The classification of k-arcs and cubic surfaces with twenty-seven lines over the field of eleven elements, D.Phil. Thesis, University of Sussex, 1984.

    Google Scholar 

  193. M. Scipioni, Sugli archi completi nei piani desarguesiani, Tesi di laurea, Uni-versity of Rome “La Sapienza”, 1990.

    Google Scholar 

  194. B. Segre, Sulle ovali nei piani lineari finiti, Atti Accad. Naz. Lincei Rend. 17 (1954), 1–2.

    MathSciNet  Google Scholar 

  195. B. Segre, Ovals in a finite projective plane, Canad. J. Math. 7 (1955), 414–416.

    Article  MathSciNet  MATH  Google Scholar 

  196. B. Segre, Curve razionali normali e k-archi negli spazi finiti, Ann. Mat. Pura Appl. 39 (1965), 367–379.

    Google Scholar 

  197. B. Segre, Sui k-archi nei piani finiti di caratteristica due, Rev. Math. Pures Appl. 2 (1957), 289–300.

    MathSciNet  Google Scholar 

  198. B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. 48 (1959), 1–97.

    Article  MathSciNet  MATH  Google Scholar 

  199. B. Segre, Lectures on Modern Geometry, Cremonese, Rome, 1961.

    Google Scholar 

  200. B. Segre, Ovali e curve σ nei piani di Galois di caratteristica due, Atti dell’ Accad. Naz. Lincei Rend. (8) 32 (1962), 785–790.

    MathSciNet  MATH  Google Scholar 

  201. B. Segre, Introduction to Galois geometries (ed. J.W.P. Hirschfeld), Atti Accad. Naz. Lincei Mem. 8 (1967), 133–236.

    Google Scholar 

  202. G. Solomon and J.J. Stiffier, Algebraically punctured cyclic codes, Inform. and Control 8 (1965), 170–179.

    Article  MATH  Google Scholar 

  203. L. Storme and P. Sziklai, Linear point sets and Rédei type k-blocking sets in PG(n, q), submitted.

    Google Scholar 

  204. L. Storme and T. Szőnyi, Caps in PG(n, q), q even, n ≥ 3, Geom. Dedicata 45 (1993), 163–169.

    Article  MathSciNet  MATH  Google Scholar 

  205. L. Storme and J.A. Thas, Complete k-arcs in PG(n, q), q even, Discrete Math. 106/107 (1992), 455–469.

    Article  MathSciNet  Google Scholar 

  206. L. Storme and J.A. Thas, M.D.S. codes and arcs in PG(n, q) with q even: an improvement of the bounds of Bruen, Thas and Blokhuis, J. Combin. Theory Ser. A 62 (1993), 139–154.

    Article  MathSciNet  MATH  Google Scholar 

  207. L. Storme and J.A. Thas, k-arcs and dual k-arcs, Discrete Math. 125 (1994), 357–370.

    Article  MathSciNet  MATH  Google Scholar 

  208. L. Storme, J.A. Thas and S.K.J. Vereecke, New upper bounds for the sizes of caps in finite projective spaces, J. Geom., to appear.

    Google Scholar 

  209. L. Storme and Z. Weiner, Minimal blocking sets in PG(n, q), n ≥ 3, Des. Codes Cryptogr. 21 (2000), 235–251.

    Article  MathSciNet  MATH  Google Scholar 

  210. P. Sziklai and T. Szőnyi, Blocking sets and algebraic curves, Rend. Circ. Mat. Palermo 51 (1998), 71–86.

    Google Scholar 

  211. T. Szőnyi, Small complete arcs in Galois planes, Geom. Dedicata 18 (1985), 161–172.

    Article  MathSciNet  Google Scholar 

  212. T. Szőnyi, Note on the order of magnitude of k for complete k-arcs in PG(2, q), Discrete Math. 66 (1987), 279–282.

    Article  MathSciNet  Google Scholar 

  213. T. Szőnyi, Arcs in cubic curves and 3-independent sets of abelian groups, Combinatorics, Eger, Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam 52 (1987), 499–508.

    Google Scholar 

  214. T. Szőnyi, Note on the existence of large minimal blocking sets in Galois planes, Combinatorica 12 (1992), 227–235.

    Article  MathSciNet  Google Scholar 

  215. T. Szőnyi, Arcs, caps, codes, and 3-independent subsets, Giornate di Geometrie Combinatorie (eds. G. Faina and G. Tallini), Università degli Studi di Perugia, Perugia, 1993, 57–80.

    Google Scholar 

  216. T. Szőnyi, Blocking sets in Desarguesian affine and projective planes, Finite Fields Appl. 3 (1997), 187–202.

    Article  MathSciNet  Google Scholar 

  217. T. Szőnyi, On the embedding of (k, p)-arcs in maximal arcs, Des. Codes Cryptogr. 18 (1999), 235–246.

    Article  MathSciNet  Google Scholar 

  218. T. Szőnyi and Z. Weiner, Small blocking sets in higher dimensions, J. Combin. Theory Ser. A, to appear.

    Google Scholar 

  219. G. Tallini, Calotte complete di S 4,q contenenti due quadriche ellittiche quali sezioni iperpiane, Rend. Mat. e Appl. 23 (1964), 108–123.

    MathSciNet  MATH  Google Scholar 

  220. J.A. Thas, Normal rational curves and k-arcs in Galois spaces, Rend. Mat. 1 (1968), 331–334.

    MathSciNet  MATH  Google Scholar 

  221. J.A. Thas, Connection between the Grassmannian G k − 1;n and the set of the k-arcs of the Galois space S n,q , Rend. Mat. 2 (1969), 121–134.

    MathSciNet  MATH  Google Scholar 

  222. J.A. Thas, Some results concerning {(q + 1)(n − 1); n}-arcs and {(q + 1)(n − 1) + 1; n}-arcs in finite projective planes of order q, J. Combin. Theory Ser. A 19 (1975), 228–232.

    Article  MathSciNet  MATH  Google Scholar 

  223. J.A. Thas, Complete arcs and algebraic curves in PG(2, q), J. Algebra 106 (1987), 451–464.

    Article  MathSciNet  MATH  Google Scholar 

  224. J. Tits, Ovoides et groupes de Suzuki, Arch. Math. 13 (1962), 187–198.

    Article  MathSciNet  Google Scholar 

  225. J.F. Voloch, On the completeness of certain plane arcs. European J. Combin. 8 (1987), 453–456.

    MathSciNet  MATH  Google Scholar 

  226. J.F. Voloch, Arcs in projective planes over prime fields, J. Geom. 38 (1990), 198–200.

    Article  MathSciNet  MATH  Google Scholar 

  227. J.F. Voloch, On the completeness of certain plane arcs II, European J. Combin. 11 (1990), 491–496.

    MathSciNet  MATH  Google Scholar 

  228. J.F. Voloch, Complete axcs in Galois planes of non-square order, Advances in Finite Geometries and Designs (eds. J.W.P. Hirschfeld, D.R. Hughes, J.A. Thas), Oxford University Press, Oxford, 1991, 401–406.

    Google Scholar 

  229. W.C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. 2 (1969), 521–560.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Sussex, Brighton, BN1 9QH, UK

    J. W. P. Hirschfeld

  2. Department of Pure Mathematics, Ghent University, Krijgslaan 281, B-9000, Gent, Belgium

    L. Storme

Authors
  1. J. W. P. Hirschfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Storme
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Eindhoven, The Netherlands

    A. Blokhuis

  2. University of Sussex, England

    J. W. P. Hirschfeld

  3. University of Augsburg, Germany

    D. Jungnickel

  4. University of Ghent, Belgium

    J. A. Thas

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Kluwer Academic Publishers

About this chapter

Cite this chapter

Hirschfeld, J.W.P., Storme, L. (2001). The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001. In: Blokhuis, A., Hirschfeld, J.W.P., Jungnickel, D., Thas, J.A. (eds) Finite Geometries. Developments in Mathematics, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0283-4_13

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0283-4_13

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7977-5

  • Online ISBN: 978-1-4613-0283-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation