Let $\mathrm{PG}(k-1,q)$ be the $(k-1)$-dimensional projective space over the finite field $\mathbb{F}_q$. An arc in $\mathrm{PG}(k-1,q)$ is a set of points with the property that any $k$ of them span the entire space. The notion of pseudo-arc generalizes that of an arc by replacing points with higher-dimensional subspaces. Constructions of pseudo-arcs can be obtained from arcs defined over extension fields; such pseudo-arcs are necessarily Desarguesian, in the sense that all their elements belong to a Desarguesian spread. In contrast, genuinely non-Desarguesian pseudo-arcs are far less understood and have previously been known only in a few sporadic cases. In this paper, we introduce a new infinite family of non-Desarguesian pseudo-arcs consisting of $(h-1)$-dimensional subspaces of $\mathrm{PG}(k-1,q)$ based on the imaginary spaces of a normal rational curve. We determine the size of the constructed pseudo-arcs explicitly and show that, by adding suitable osculating spaces of a normal rational curve defined over a subgeometry, we obtain pseudo-arcs of size $O(q^h)$. As $q$ grows, these sizes asymptotically attain the classical upper bound for pseudo-arcs established in 1971 by J.~A.~Thas, thereby showing that this bound is essentially sharp also in the non-Desarguesian setting. We further investigate the interaction between these new pseudo-arcs and quadrics. While Desarguesian pseudo-arcs from normal rational curve are complete intersections of quadrics, we prove that the new pseudo-arcs are not contained in any quadric of the ambient projective space. Finally, we translate our geometric results into coding theory. We show that the new pseudo-arcs correspond precisely to recent families of additive MDS codes introduced via a polynomial framework. As a consequence of their non-Desarguesian nature, we prove that these codes are not equivalent to linear MDS codes.

翻译：设 $\mathrm{PG}(k-1,q)$ 为有限域 $\mathbb{F}_q$ 上的 $(k-1)$ 维射影空间。$\mathrm{PG}(k-1,q)$ 中的弧是指满足如下性质的点的集合：其中任意 $k$ 个点张成整个空间。伪弧的概念通过将点替换为更高维子空间推广了弧的概念。伪弧的构造可以从定义在扩域上的弧获得；此类伪弧必然是笛沙格的，即其所有元素都属于一个笛沙格铺砌。相比之下，真正的非笛沙格伪弧远未得到充分理解，此前仅在少数零星情形中已知。本文中，我们基于正规有理曲线的虚空间，引入了一族新的由 $\mathrm{PG}(k-1,q)$ 中 $(h-1)$ 维子空间构成的非笛沙格伪弧。我们显式确定了所构造伪弧的规模，并证明通过添加定义在子几何上的正规有理曲线的适当密切空间，我们可以获得规模为 $O(q^h)$ 的伪弧。随着 $q$ 增大，这些规模渐近地达到了 J.~A.~Thas 于 1971 年建立的伪弧经典上界，从而表明该上界在非笛沙格情形下本质上也是紧的。我们进一步研究了这些新伪弧与二次曲面之间的相互作用。虽然来自正规有理曲线的笛沙格伪弧是二次曲面的完全交，但我们证明了新的伪弧不包含在环境射影空间的任何二次曲面中。最后，我们将几何结果转化为编码理论。我们证明这些新伪弧恰好对应于近期通过多项式框架引入的加法 MDS 码族。作为其非笛沙格性质的一个推论，我们证明了这些码不等价于线性 MDS 码。