Fundamental Notions of Projective and Scale-Translation-Invariant Metrics in Coding Theory

TL;DR

This paper develops the foundational theory of projective metrics in coding theory, exploring their properties, classifications, and connections to other metrics, and also examines scale-translation-invariant metrics.

Contribution

It introduces the fundamental theory of projective metrics, including key properties, equivalence, isometries, and their relations to classical metrics and matroids.

Findings

Characterization of projective metrics and their properties

Connections between projective and Hamming metrics

Bounds and embeddings related to scale-translation-invariant metrics

Abstract

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.