When Arcs Extend Uniquely: A Higher-Dimensional Generalization of Barlotti's Result

In this short communication, we generalize a classical result of Barlotti concerning the unique extendability of arcs in the projective plane to higher-dimensional projective spaces. Specifically, we show that for integers $k\ge 3$, $s\ge 0$, and prime power $q$, any $(n,k+s-1)$-arc in PG$(k-1,q)$ of size $n=(s+1)(q+1)+k-3$ admits a unique extension to a maximal arc, provided $s+2\mid q$ and $s<q-2$. This result extends the classical characterizations of maximal arcs in PG$(2,q)$ and connects naturally to the theory of A${}^s$MDS codes. Our findings establish conditions under which linear codes of given dimension and Singleton defect can be uniquely extended to maximal-length projective codes.