Abstract—In the projective plane PG(2, q), a subset S of a conic C is said to be almost complete if it can be extended to a larger arc in PG(2, q) only by the points of C \ S and by the nucleus of C when q is even. We obtain new upper bounds on the smallest size t(q) of an almost complete subset of a conic, in particular,$$t(q) < \sqrt {q(3lnq + lnlnq + ln3)} + \sqrt {\frac{q}{{3\ln q}}} + 4 \sim \sqrt {3q\ln q} ,t(q) < 1.835\sqrt {q\ln q.} $$The new bounds are used to extend the set of pairs (N, q) for which it is proved that every normal rational curve in the projective space PG(N, q) is a complete (q+1)-arc, or equivalently, that no [q+1,N+1, q−N+1]_q generalized doubly-extended Reed–Solomon code can be extended to a [q + 2,N + 1, q − N + 2]_q maximum distance separable code.

中文翻译：

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PG（2，q）中圆锥的几乎完整子集的最小大小和Reed-Solomon码的可扩展性

摘要-在投影平面PG（2，q）中，如果仅能通过C \ S的点将圆锥C的子集S扩展到PG（2，q）中的较大弧度，则称圆锥C的子集几乎是完整的。当q为偶数时，由C的核组成。我们在圆锥的几乎完整子集的最小尺寸t（q）上获得新的上限，特别是$$ t（q）<\ sqrt {q（3lnq + lnlnq + ln3）} + \ sqrt {\ frac {q} {{3 \ ln q}}} + 4 \ sim \ sqrt {3q \ ln q}，t（q）<1.835 \ sqrt {q \ ln q。} $$新边界用于扩展对对（N，q），证明射影空间PG（N，q）中的每个法向有理曲线都是完整的（q +1）弧，或者等效地，没有[ q +1，N +1，q-N +1] _q广义双扩展Reed–Solomon码可以扩展为[ q + 2，N + 1，q - N + 2] _q最大距离可分离码。