We show that extended cyclic codes over \(\mathbb {F}_q\) with parameters \([q+2,3,q]\), \(q=2^m\), determine regular hyperovals. We also show that extended cyclic codes with parameters \([qt-q+t,3,qt-q]\), \(1<t<q\), q is a power of t, determine (cyclic) Denniston maximal arcs. Similarly, cyclic codes with parameters \([q^2+1,4,q^2-q]\) are equivalent to ovoid codes obtained from elliptic quadrics in PG(3, q). Finally, we give simple presentations of Denniston maximal arcs in PG(2, q) and elliptic quadrics in PG(3, q).

中文翻译：

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扩展循环码、最大弧和卵形

我们展示了\(\mathbb {F}_q\) 上带有参数\([q+2,3,q]\) , \(q=2^m\) 的扩展循环码确定了规则的超椭圆。我们还展示了具有参数\([qt-q+t,3,qt-q]\) , \(1<t<q\) 的扩展循环码，q是t的幂，确定（循环）丹尼斯顿最大值弧线。类似地，参数为\([q^2+1,4,q^2-q]\) 的循环码等价于从PG (3,  q ) 中的椭圆二次方程得到的卵形码。最后，我们给出丹尼斯顿最大弧的简单介绍PG（2，  q）和椭圆形二次曲面PG(3,  q )。