The Geometry of $(t\mod{q})$-arcs

TL;DR

This paper introduces a geometric method to construct strong non-lifted $(t ext{ mod }q)$-arcs in projective spaces, providing explicit examples and an infinite family for certain parameters, advancing combinatorial geometry.

Contribution

It presents the first geometric construction of specific non-lifted $(t ext{ mod }q)$-arcs and generalizes to an infinite family for all odd prime powers and dimensions.

Findings

Constructed three specific strong non-lifted $(3 ext{ mod }5)$-arcs in $ ext{PG}(3,5)$.

Developed an infinite family of non-lifted, strong $(t ext{ mod }q)$-arcs for $t=(q+1)/2$ in $ ext{PG}(r,q)$.

Provided explicit sizes and geometric descriptions of these arcs.

Abstract

In this paper, we give a geometric construction of the three strong non-lifted $(3\mod{5})$-arcs in $\operatorname{PG}(3,5)$ of respective sizes 128, 143, and 168, and construct an infinite family of non-lifted, strong $(t\mod{q})$-arcs in $\operatorname{PG}(r,q)$ with $t=(q+1)/2$ for all $r\ge3$ and all odd prime powers $q$.