Exact Decoding Probability of Sparse Random Linear Network Coding for Reliable Multicast

TL;DR

This paper derives an exact formula for the decoding success probability of sparse random linear network coding (SRLNC), addressing a gap where previous expressions were only asymptotic or approximate, thus enabling precise reliability analysis.

Contribution

The paper introduces a novel criterion for a vector's inclusion in a subspace and constructs a basis to derive an exact decoding probability expression for SRLNC.

Findings

Exact decoding success probability formula derived

Demonstrated the formula's accuracy with a simple example

Provides a foundation for precise reliability analysis in SRLNC

Abstract

Sparse random linear network coding (SRLNC) used as a class of erasure codes to ensure the reliability of multicast communications has been widely investigated. However, an exact expression for the decoding success probability of SRLNC is still unknown, and existing expressions are either asymptotic or approximate. In this paper, we derive an exact expression for the decoding success probability of SRLNC. The key to achieving this is to propose a criterion that a vector is contained in a subspace. To obtain this criterion, we construct a basis of a subspace, with respect to this basis, the coordinates of a vector are known, based on a maximal linearly independent set of the columns of a matrix. The exactness and the computation of the derived expression are demonstrated by a simple example.