2025/05/09 by Riccardi, Gabor, Hugo Sauerbier Couvée, Couvée, Hugo Sauerbier · 1 citation
Computer Science · Mathematics · #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Fixed Point Theorems Analysis #Graph Labeling and Dimension Problems #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2505.06388
openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Projective metrics on vector spaces over finite fields, introduced by Gabidulin and Simonis in 1997, generalize classical metrics in coding theory like the Hamming metric, rank metric, and combinatorial metrics. While these specific metrics have been thoroughly investigated, the overarching theory of projective metrics has remained underdeveloped since their introduction. In this paper, we present and develop the foundational theory of projective metrics, establishing several elementary key results on their characterizing properties, equivalence classes, isometries, constructions, connections with the Hamming metric, associated matroids, sphere sizes and Singleton-like bounds. Furthermore, some general aspects of scale-translation-invariant metrics are examined, with particular focus on their embeddings into larger projective metric spaces.