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.