Fountain Codes and Invertible Matrices

TL;DR

This paper explores the properties of Fountain code encoding matrices, demonstrating their permutation and group structures, and introduces a special matrix that more effectively reduces entropy compared to traditional distributions, supported by experimental results.

Contribution

It establishes that Fountain code encoding matrices form a permutation group and introduces a novel matrix that enhances entropy reduction over existing distributions.

Findings

Encoding matrices induce permutations.

Encoding matrices form a group under multiplication.

A new matrix achieves better entropy reduction than Ideal Soliton-based matrices.

Abstract

This paper deals with Fountain codes, and especially with their encoding matrices, which are required here to be invertible. A result is stated that an encoding matrix induces a permutation. Also, a result is that encoding matrices form a group with multiplication operation. An encoding is a transformation, which reduces the entropy of an initially high-entropy input vector. A special encoding matrix, with which the entropy reduction is more effective than with matrices created by the Ideal Soliton distribution is formed. Experimental results with entropy reduction are shown.