The values \(\sqrt {2q}\) and \(\log _{2}q\):\, their relationship with \(k\)-arcs. (Q2715986)
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scientific article; zbMATH DE number 1600955
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The values \(\sqrt {2q}\) and \(\log _{2}q\):\, their relationship with \(k\)-arcs. |
scientific article; zbMATH DE number 1600955 |
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20 July 2005
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\(k\)-arc
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complete \(k\)-arc
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secant
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0.8069547
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0.79338694
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0.7915307
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0.7862175
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0.7795471
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The values \(\sqrt {2q}\) and \(\log _{2}q\):\, their relationship with \(k\)-arcs. (English)
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Let \(K = \{P_1,\dots ,P_k\}\) be a \(k\)-arc in a projective plane of order \(q\). Denote by \(a_i\) the number of points that are covered by secants of \(\{P_1,\dots ,P_i\}\), but not by secants of \(\{P_1,\dots ,P_{i-1}\}\). Then \(b_i \leq a_i \leq c_i\), where \(b_i = qi - q - i^3/2 + 3i^2 - 13i/2 + 5\) and \(c_i = qi - q - i^2 +3i -1\). These estimates are used to show that if \(k = [\sqrt {2q}] + 2\), then secants of \(K\) fill about half the plane. One also gets an estimate of \(k\) for the case when \(K\) is complete. In the case \(k=[\sqrt {2q}] + 2\) the author estimates the number of \(k\)-arcs needed to cover the whole plane.
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