On the minimum number of points covered by a set of lines in \(PG(2, q)\)

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• 作者：Eun Ju Cheon ; Seon Jeong Kim
• 关键词：Projective plane ; Rational point ; Arc ; Hyperoval ; Conic ; Largest arc ; 05B25 ; 51E21 ; 51E15 ; 14G05
• 刊名：Designs, Codes and Cryptography
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：74
• 期：1
• 页码：59-74
• 全文大小：252 KB
• 参考文献：1. Ball S., Hirschfeld J.W.P.: Bounds on \((n, r)\) -arcs and their application to linear codes. Finite Fields Appl. 11, 326-36 (2005).
  2. Blokhuis A.: Extremal problems in finite geometries. In: Frankl P., Füredi Z., Katona G., Miklós D. (eds.) Extremal Problems for Finite Sets, vol. 3, pp. 111-35. Bolyai Society Mathematical Studies, Budapest (1994).
  3. Blokhuis A., Bruen A.A.: The minimal number of lines intersected by a set of \(q+2\) points, blocking sets, and intersecting circles. J. Comb. Theory Ser. A 50(2), 308-15 (1989).
  4. Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998).
  5. Segre B.: Le geometrie di Galois. Ann. Mat. Pura Appl. 48, 1-6 (1959).
  6. Weiner Z., Sz?nyi T.: On the stability of the sets of even type. http://www.cs.elte.hu/~weiner. Accessed 27 June 2013
  
• 作者单位：Eun Ju Cheon (1)
  Seon Jeong Kim (1)
  
  1. Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

Segre (Ann Mat Pura Appl 48:1-6, 1959) mentioned that the number \(N\) of points on a curve which splits into \(k\) distinct lines on the projective plane over a finite field of order \(q\) satisfies \(kq - \frac{k(k-3)}{2} \le N \le kq+1.\) We see that the upper bound is satisfactory, but the lower one is not for \(k\ge q+2\) [resp. \(k\ge q+3\) ] if \(q\) is odd [resp. even]. We consider the minimum number \(m_q(k)\) of points on such a curve of degree \(k\) , and obtain the complete sequence \(\{m_q(k) \mid 0 \le k\le q^2+q+1\}\) for every prime power \(q\le 8\) .