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

TL;DR

This paper generalizes Barlotti's classical result on the unique extension of arcs in projective planes to higher-dimensional projective spaces, establishing conditions for unique extensions of certain arcs and their connection to maximal codes.

Contribution

It introduces a higher-dimensional generalization of Barlotti's result, providing new conditions for unique arc extensions and linking to the theory of A$^s$MDS codes.

Findings

Unique extension conditions for higher-dimensional arcs

Connection between arc extensions and A$^s$MDS codes

Extension of classical projective plane results to higher dimensions

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.