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Structural Redundancy in Subspace Network Coding via Atomic Decompositions

David Ramirez, Elvis Cabrera, Jyrko Correa-Morris

Abstract

Random linear network coding (RLNC) provides a powerful framework for non-coherent communication, where reliable transmission requires correcting errors and erasures induced by network mixing and motivates the use of subspace codes. In this work, we introduce an atomic perspective on subspace coding by formalizing the notion of minimal atomic decompositions in the lattice L(V ) of subspaces of a finite-dimensional vector space over a finite field. We study the function N that assigns to each subspace the number of its minimal atomic decompositions and establish its key structural properties. Leveraging N, we define a new distance metric on L(V ) that refines classical subspace comparisons by capturing atomic-level overlap. We then introduce the Atomic Operator Channel, a transmission model for RLNC in which codewords are conveyed through atomic decompositions and corruption is modeled via atomic insertions and erasures. Within this framework, we prove a minimum-distance decoding guarantee for the induced metric. In the constant-dimension setting, we show that the classical unique-decodability condition under the subspace distance remains sufficient for unique decoding under the atomic metric.

Structural Redundancy in Subspace Network Coding via Atomic Decompositions

Abstract

Random linear network coding (RLNC) provides a powerful framework for non-coherent communication, where reliable transmission requires correcting errors and erasures induced by network mixing and motivates the use of subspace codes. In this work, we introduce an atomic perspective on subspace coding by formalizing the notion of minimal atomic decompositions in the lattice L(V ) of subspaces of a finite-dimensional vector space over a finite field. We study the function N that assigns to each subspace the number of its minimal atomic decompositions and establish its key structural properties. Leveraging N, we define a new distance metric on L(V ) that refines classical subspace comparisons by capturing atomic-level overlap. We then introduce the Atomic Operator Channel, a transmission model for RLNC in which codewords are conveyed through atomic decompositions and corruption is modeled via atomic insertions and erasures. Within this framework, we prove a minimum-distance decoding guarantee for the induced metric. In the constant-dimension setting, we show that the classical unique-decodability condition under the subspace distance remains sufficient for unique decoding under the atomic metric.
Paper Structure (22 sections, 10 theorems, 69 equations)

This paper contains 22 sections, 10 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Minimal Atomic Decompositions in $\mathcal{L}(V)$
  4. Atomic Decompositions
  5. Characterization of Minimal Atomic Decompositions
  6. Counting Minimal Atomic Decompositions
  7. The Atomic Operator Channel
  8. Atomic Representation of Transmissions
  9. Redundancy and Structural Robustness
  10. The Atomic Operator Channel
  11. The $\mathfrak{N}$-Induced Distance
  12. Supermodularity of $\mathfrak N$
  13. Definition of the $\mathfrak N$-Induced Distance
  14. Metric Properties
  15. Computational complexity of evaluating $d_{\mathfrak N}$
  16. ...and 7 more sections

Key Result

lemma 1

A subset $\mathfrak a_S=\{A_1,A_2,\dots,A_k\}\subset \mathcal{A}$ is a minimal atomic decomposition of the subspace $S$ if and only if every choice of vectors $\{v_1,v_2,\ldots,v_k\}$ with $v_i\in A_i$ and $v_i\neq 0$ forms a basis of $S$.

Theorems & Definitions (26)

  • definition 1
  • lemma 1
  • proof
  • definition 2
  • lemma 2
  • proof
  • theorem 1
  • proof
  • corollary 1
  • proof
  • ...and 16 more