Families of Optimal Binary Non-MDS Erasure Codes

TL;DR

This paper defines and constructs families of optimal binary non-MDS erasure codes with higher decoding probabilities than random linear codes over GF(2), approaching the performance of codes over GF(4) for small k.

Contribution

It introduces a new class of optimal binary erasure codes and an algorithm to find them using hill climbing on Balanced XOR codes.

Findings

Codes outperform random linear codes over GF(2)

Decoding probability close to GF(4) codes for small k

Algorithm effectively finds high-performance code families

Abstract

We introduce a definition for \emph{Families of Optimal Binary Non-MDS Erasure Codes} for $[n, k]$ codes over $GF(2)$, and propose an algorithm for finding those families by using hill climbing techniques over Balanced XOR codes. Due to the hill climbing search, those families of codes have always better decoding probability than the codes generated in a typical Random Linear Network Coding scenario, i.e., random linear codes. We also show a surprising result that for small values of $k$, the decoding probability of our codes in $GF(2)$ is very close to the decoding probability of the codes obtained by Random Linear Network Coding but in the higher finite field $GF(4)$.