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

Gabor Riccardi, Hugo Sauerbier Couvée

TL;DR

We address unifying projective metrics on finite-vector spaces by proving that a projective metric is exactly an integral convex scale- and translation-invariant metric and can be realized as a quotient of a Hamming metric via a family of 1-dimensional subspaces. We construct a bijection between isomorphism classes of projective metrics on a space $V$ and the set of Hamming-equivalence classes of parent codes in ${\mathbb F}_q^N$, and show that the projective isometry group equals the stabilizer of the corresponding parent code. We prove existence of embeddings of any finitely-valued scale-invariant metric into a projective metric on a larger space and relate sphere sizes to coset distributions through extended spanning families and an associated matroid, yielding a Singleton-type bound. The work provides extensive examples, a Sage implementation, and sets a foundation for connections among coding theory, matroid theory, graph theory, and finite geometry.

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.

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

TL;DR

We address unifying projective metrics on finite-vector spaces by proving that a projective metric is exactly an integral convex scale- and translation-invariant metric and can be realized as a quotient of a Hamming metric via a family of 1-dimensional subspaces. We construct a bijection between isomorphism classes of projective metrics on a space and the set of Hamming-equivalence classes of parent codes in , and show that the projective isometry group equals the stabilizer of the corresponding parent code. We prove existence of embeddings of any finitely-valued scale-invariant metric into a projective metric on a larger space and relate sphere sizes to coset distributions through extended spanning families and an associated matroid, yielding a Singleton-type bound. The work provides extensive examples, a Sage implementation, and sets a foundation for connections among coding theory, matroid theory, graph theory, and finite geometry.

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.
Paper Structure (33 sections, 46 theorems, 149 equations, 1 figure)

This paper contains 33 sections, 46 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Structure of the paper
  4. Preliminaries on metrics
  5. Metrics and weight functions
  6. Weighted vector spaces and contractions
  7. Introduction to Projective Metrics
  8. Why projective metrics?
  9. Projective weight functions and projective metrics
  10. Examples
  11. Projective metrics as quotients
  12. Parent codes and parent functions
  13. Projective isometries
  14. Decoding of the parent code and projective weight algorithms.
  15. Embeddings of scale-translation-invariant metrics into projective metrics
  16. ...and 18 more sections

Key Result

Theorem 4

The quotient weight $\operatorname{wt}_{\operatorname{quot},\varphi}$ with respect to a surjective linear map $\varphi: (X, \operatorname{wt}_X) \twoheadrightarrow Y$ satisfies the following:

Figures (1)

  • Figure 1: Cayley graph associated to the phase rotation metric on $\mathbb{F}_2^4$. Indicated are shortest paths from $(0,0,0,0)$ to $(1,1,0,1)$ in the phase rotation metric (blue) and in the Hamming metric (red).

Theorems & Definitions (130)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark
  • Theorem 4
  • Corollary 5
  • Definition 5
  • Proposition 6
  • Proposition 7
  • Corollary 7
  • ...and 120 more