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

Tim L. Alderson

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.

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 , , and prime power , 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 and . This result extends the classical characterizations of maximal arcs in PG\((2,q)\) and connects naturally to the theory of AMDS codes. Our findings establish conditions under which linear codes of given dimension and Singleton defect can be uniquely extended to maximal-length projective codes.

Paper Structure

This paper contains 10 sections, 8 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Projective Geometry and Quotient Geometries
  3. Arcs and Multi-Arcs in Projective Space
  4. Maximal Arcs and the Function $m^s(k, q)$
  5. Maximal Arcs in the Plane
  6. Maximal Arcs in Higher Dimensions
  7. Interpretation in Terms of Linear Codes
  8. Conclusion

Key Result

Lemma 1

Theorems & Definitions (11)

  • Lemma 1: MR0083141,MR1466573, MR0239991
  • Lemma 2: MR0083141
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 1
  • Definition 1
  • Corollary 2
  • Corollary 3
  • ...and 1 more