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On the uniqueness of \((q+1)\)-arcs of \(PG(5,q),\quad q=2^ h,\quad h\geq{} 4\). (English) Zbl 0735.51012

It is well known that, if \(q=2^ h\), with \(h\geq 3\), then every \((q+1)\)- arc in \(PG(4,q)\) is a normal rational curve and that every \((q+1)\)-arc in \(PG(3,q)\) is projectively equivalent to \(K(r)=\{(1,t,t^ r,t^{r+1})\mid t\in F\}\cup\{(0,0,0,1)\}\) for some \(r=2^ n\), \((n,h)=1\). Inspired by these results, the authors prove that: if \(q=2^ h\), \(h\geq 4\), then (a) every \((q+1)\)-arc in \(PG(5,q)\) is a normal rational curve; (b) in \(PG(4,q)\), every \(q\)-arc is contained in a unique \((q+1)\)-arc; (c) in \(PG(3,q)\), every \((q-1)\)-arc is contained in a unique \((q+1)\)-arc.
Reviewer: G.Faina

MSC:

51E21 Blocking sets, ovals, \(k\)-arcs
51E20 Combinatorial structures in finite projective spaces
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References:

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