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Erschienen in:
01.01.2015
On the minimum number of points covered by a set of lines in \(PG(2, q)\)
Erschienen in: Designs, Codes and Cryptography | Ausgabe 1/2015
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Abstract
Segre (Ann Mat Pura Appl 48:1–96, 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 \(q\le 8\).