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

Abstract: 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.