Fast Decoding of Lifted Interleaved Linearized Reed-Solomon Codes for Multishot Network Coding

TL;DR

This paper introduces efficient decoding methods for lifted interleaved linearized Reed-Solomon (LILRS) codes, enhancing error correction in multishot network coding by increasing decoding region and reducing overhead.

Contribution

It presents new construction and decoding schemes for LILRS codes, allowing correction of insertions and deletions beyond half the minimum distance with low failure probability.

Findings

Decoding schemes can correct errors beyond half the minimum distance.

Proposed decoders have very small failure probabilities in most scenarios.

Monte Carlo simulations confirm the tightness of failure probability bounds.

Abstract

Mart{\'\i}nez-Pe{\~n}as and Kschischang (IEEE Trans.\ Inf.\ Theory, 2019) proposed lifted linearized Reed--Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode \ac{LILRS} codes. Compared to the construction by Mart{\'\i}nez-Pe{\~n}as--Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for \ac{LILRS} that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique {\LOlike} decoder for \ac{LILRS} codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations.