Maruta, Tatsuya; Kaneta, Hitoshi On the uniqueness of \((q+1)\)-arcs of \(PG(5,q),\quad q=2^ h,\quad h\geq{} 4\). (English) Zbl 0735.51012 Math. Proc. Camb. Philos. Soc. 110, No. 1, 91-94 (1991). 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 Cited in 3 Documents MSC: 51E21 Blocking sets, ovals, \(k\)-arcs 51E20 Combinatorial structures in finite projective spaces Keywords:k-arcs; rational curve PDFBibTeX XMLCite \textit{T. Maruta} and \textit{H. Kaneta}, Math. Proc. Camb. Philos. Soc. 110, No. 1, 91--94 (1991; Zbl 0735.51012) Full Text: DOI References: [1] DOI: 10.1109/18.45291 · Zbl 0695.94010 · doi:10.1109/18.45291 [2] Kaneta, Simon Stevin 63 pp 363– (1989) [3] Kaneta, Math. Proc. Cambridge Philos. Soc. 105 pp 459– (1989) · Zbl 0688.51007 · doi:10.1017/S0305004100077823 [4] DOI: 10.1007/BF01393742 · Zbl 0654.94014 · doi:10.1007/BF01393742 [5] Hirschfeld, Projective Geometries over Finite Fields (1979) · Zbl 0418.51002 [6] DOI: 10.1016/0012-365X(84)90180-8 · Zbl 0535.51016 · doi:10.1016/0012-365X(84)90180-8 [7] DOI: 10.1007/BF00147659 · Zbl 0503.51011 · doi:10.1007/BF00147659 [8] Hirschfeld, Finite Projective Spaces of Three Dimensions (1985) · Zbl 0574.51001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.