On the sharpness of a theorem of B. segre

Abstract

The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s 2 by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q 3) over GF(q), and on a representation of GF(q)3 in GF(q 3)3 indicated in Jamison [4].

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References

  1. [1]

    R. H. Bruck, Quadratic extension of cyclic planes,Proc. Symp. in Appl. Math. (X), 1960, (ed. byR. Bellman andM. Hall Jr.), 15–44.

  2. [2]

    M. Hall, Jr., Cyclic projective planes,Duke Math. Journal 14 (1947), 1079–1090.

    MATH  Article  Google Scholar 

  3. [3]

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

    Google Scholar 

  4. [4]

    R. E. Jamison, Covering finite fields with cosets of subspaces,Journ. Comb. Th. (A) 22 (1977), 253–266.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    B. C. Kestenband, Unital intersections in finite projective planes,Geometriae Ded. 11 (1981), 107–117.

    MATH  MathSciNet  Google Scholar 

  6. [6]

    R. Lidl andH. Niederreiter,Finite Fields, Encyclopaedia of Math. 20, Addison-Wesley, 1983.

  7. [7]

    B. Segre, Introduction to Galois geometries, (ed. byJ. W. P. Hirschfeld),Memorie Accad. Naz. Lincei (VIII),5 (1967), 133–236.

    MathSciNet  Google Scholar 

  8. [8]

    G. Seroussi andA. Lempel, Factorization of symmetric matrices and trace-orthogonal bases in finite fields,SIAM J. Computing 9 (1980), 758–767.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    J. Singer, A theorem in finite projective geometry and some applications to number theory,Trans. Amer. Math. Soc. 43 (1938), 377–385.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    J. A. Thas, Elementary proofs of two fundamental theorem of B. Segre without using the Hasse—Weil theorem,Journ. of Comb. Th. (A) 34 (1983), 381–348.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    P. Yff, On subplane partition of a finite projective plane,Journal of Comb. Th. (A) 22 (1977), 118–122.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

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

    MathSciNet  Google Scholar 

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Boros, E., Szőnyi, T. On the sharpness of a theorem of B. segre. Combinatorica 6, 261–268 (1986). https://doi.org/10.1007/BF02579386

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AMS subject classification (1980)

  • 05 B 25
  • 51 E 15
  • 51 E 20