Baer Subplanes, Ovals and Unitals

  • E. F. AssmusJr.
  • J. D. Key
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 20)


1. Introduction. In this paper we examine some of the types of codewords that can occur in codes associated with finite projective and affine planes, exploring further the notions that were introduced in [1]. There we defined the hull, \( H_p \left( D \right) \), of a design \( \left( D \right) \) over a finite field \(F_p\), where p is a prime that divides the order n of the design: if \( C_p \left( D \right) \) denotes the code of \( \left( D \right) \) over F p , defined to be the space spanned by the characteristic functions of the blocks of \( \left( D \right) \), then \( H_p \left( D \right)\; = \;C_p \left( D \right)\; \cap \;C_P \left( D \right)^ \bot \), where X denotes the subspace orthogonal to X with respect to the standard inner product. In the case where \( \left( D \right) \) is a finite projective plane П, and π any affine part of П, we showed that affine planes obtained by “derivation” from π (see [1]) could be obtained from minimal-weight vectors of \(H_p(\pi)^{\bot}\). Since \( {{H}_{p}}{{(\pi )}^{ \bot }} \) is the image of the natural projection of \(H_p(\Pi)^{\bot}\) we were led to examine the nature of codewords of the code \(H_p(\Pi)^{\bot}\), and, in particular, those codewords that could give rise to minimal-weight (i.e. weight-n) vectors in \(H_p(\pi)^{\bot}\) and \(C_p(\pi)\). The support of such a vector must form either a line or a blocking set for П; examples of these, other than lines, are Bær subplanes meeting the line at infinity for π in a line segment, or, in the even-order case, ovals meeting that line in two points.


Projective Plane Translation Plane Affine Plane Absolute Point Desarguesian Plane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    E.F. Assmus, Jr. and J.D. Key, Affine and projective planes, Discrete Math., Special Coding Theory Issue (to appear).Google Scholar
  2. [2]
    Bhaskar Bagchi and N.S. Narasimha Sastry, Even order inversive planes, generalized quadrangles and codes, Geometriae Dedicata, 22 (1987), pp. 137–147.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    A.A. Bruen and J.W.P. Hirschfeld, Intersections in projective space I, Combinatorics, Math. Z., 193 (1986), pp. 215–225.MathSciNetzbMATHGoogle Scholar
  4. [4]
    A.A. Bruen and U. Ott, On thep-rank of incidence matrices and a question of E.S. Lander, (preprint).Google Scholar
  5. [5]
    J.C. Fisher, J.W.P. Hirschfeld and J.A. Thas, Complete arcs in planes of square order, Annals of Discrete Math., 30 (1986), pp. 243–250.MathSciNetGoogle Scholar
  6. [6]
    J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Oxford, 1979.zbMATHGoogle Scholar
  7. [7]
    D.R. Hughes and F.C. Piper, Projective Planes, Springer Graduate Texts in Mathematics (1973).zbMATHGoogle Scholar
  8. [8]
    D.R. Hughes and F.C. Piper, Design Theory, Cambridge University Press, 1985.zbMATHCrossRefGoogle Scholar
  9. [9]
    Barbu C. Kestenband, Unital intersections in finite projective planes, Geometriae Dedicata, 11 (1981), pp. 107–117.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    E.S. Lander, Symmetric Designs: an Algebraic Approach, London Mathematical Society Lecture Notes #74, Cambridge University Press, 1983.zbMATHCrossRefGoogle Scholar
  11. [11]
    Rudolf Metz, On a class of unitals, Geometriae Dedicata, 8 (1979), pp. 125–126.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    T.G. Ostrom, Finite Translation Planes, Lecture Notes in Mathematics, 158, Springer, 1970.zbMATHGoogle Scholar
  13. [13]
    H. Sachar, The F p span of the incidence matrix of a finite projective plane, Geometriae Dedicata, 8 (1979), pp. 407–415.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • E. F. AssmusJr.
    • 1
  • J. D. Key
    • 2
    • 3
  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA
  2. 2.Department of Mathematical Sciences, Martin HallClemson UniversityClemsonUSA
  3. 3.Department of MathematicsUniversity of BirminghamBirminghamUK

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