On Arcs and Quadrics
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- 关键词：2010 Mathematics Subject Classification ; 51E21 ; 15A03 ; 94B05 ; 05B35
- 刊名：Lecture Notes in Computer Science
- 出版年：2016
- 出版时间：2016
- 年：2016
- 卷：10064
- 期：1
- 页码：95-102
- 丛书名：Arithmetic of Finite Fields
- ISBN：978-3-319-55227-9
- 卷排序：10064
文摘
An arc is a set of points of the \((k-1)\)-dimensional projective space over the finite field with q elements \({\mathbb F}_q\), in which every k-subset spans the space. In this article, we firstly review Glynn’s construction of large arcs which are contained in the intersection of quadrics. Then, for q odd, we construct a series of matrices \(\mathrm {F}_n\), where n is a non-negative integer and \(n \leqslant |G|-k-1\), which depend on a small arc G. We prove that if G can be extended to a large arc S of size \(q+2k-|G|+n-2\) then, for each vector v of weight three in the column space of \(\mathrm {F}_n\), there is a quadric \(\psi _v\) containing \(S \setminus G\). This theorem is then used to deduce conditions for the existence of quadrics containing all the vectors of S.