On pseudo-arcs from normal rational curve and additive MDS codes

This is the authors' abstract. We don't add key points for this paper.

Let normal upper P normal upper G left parenthesis k minus 1 comma q right parenthesis P G ( k − 1 , q ) PG(k-1,q) be the left parenthesis k minus 1 right parenthesis ( k − 1 ) $(k-1)$ -dimensional projective space over the finite field double struck upper F Subscript q F q Fq . An arc in normal upper P normal upper G left parenthesis k minus 1 comma q right parenthesis P G ( k − 1 , q ) PG(k-1,q) is a set of points with the property that any k of them span the entire space. The notion of a pseudo-arc generalises 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 left parenthesis h minus 1 right parenthesis ( h − 1 ) $(h-1)$ -dimensional projective subspaces of normal upper P normal upper G left parenthesis h k minus 1 comma q right parenthesis P G ( h k − 1 , q ) PG(hk-1,q) based on the imaginary spaces of a normal rational curve. We determine the size of the constructed pseudo-arcs explicitly