Structural Redundancy in Subspace Network Coding via Atomic Decompositions

TL;DR

This paper introduces an atomic perspective on subspace coding for network communication, defining a new distance metric and decoding guarantees based on atomic decompositions, enhancing understanding of error correction in RLNC.

Contribution

It formalizes minimal atomic decompositions in subspace codes, introduces a new atomic-based distance metric, and analyzes decoding guarantees within this novel framework.

Findings

Defined the function N for minimal atomic decompositions

Introduced the Atomic Operator Channel model

Proved decoding guarantees under the atomic metric

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.