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.
