List and Probabilistic Unique Decoding of Folded Subspace Codes

TL;DR

This paper introduces folded subspace codes for noncoherent network coding, with an efficient decoding algorithm that corrects insertions and deletions beyond the unique decoding radius, offering probabilistic unique decoding with high probability.

Contribution

It presents a new class of folded subspace codes and an interpolation-based decoding algorithm capable of correcting errors beyond the traditional radius, including probabilistic unique decoding.

Findings

Decoding algorithm corrects errors beyond the unique decoding radius.

Provides an upper bound on list size and failure probability.

Enables probabilistic unique decoding up to the list decoding radius.

Abstract

A new class of folded subspace codes for noncoherent network coding is presented. The codes can correct insertions and deletions beyond the unique decoding radius for any code rate $R\in[0,1]$. An efficient interpolation-based decoding algorithm for this code construction is given which allows to correct insertions and deletions up to the normalized radius $s(1-((1/h+h)/(h-s+1))R)$, where $h$ is the folding parameter and $s\leq h$ is a decoding parameter. The algorithm serves as a list decoder or as a probabilistic unique decoder that outputs a unique solution with high probability. An upper bound on the average list size of (folded) subspace codes and on the decoding failure probability is derived. A major benefit of the decoding scheme is that it enables probabilistic unique decoding up to the list decoding radius.