Basis-Finding Algorithm for Decoding Fountain Codes for DNA-Based Data Storage

TL;DR

This paper introduces a basis-finding decoding algorithm for fountain codes used in DNA data storage, improving error correction efficiency by identifying reliable basis elements, with theoretical and simulation validation.

Contribution

The paper presents a novel basis-finding algorithm (BFA) for fountain code decoding that handles errors effectively and has polynomial time complexity, with theoretical bounds and superior performance over belief propagation.

Findings

BFA achieves lower FER than BP in simulations.

FER decreases as basis element weight increases.

Theoretical bounds for FER of BFA are derived.

Abstract

In this paper, we consider the decoding of fountain codes where the received symbols may have errors. It is motivated by the application of fountain codes in DNA-based data storage systems where the inner code decoding, which generally has undetectable errors, is performed before the outer fountain code decoding. We propose a novel and efficient decoding algorithm, namely basis-finding algorithm (BFA), followed by three implementations. The key idea of the BFA is to find a basis of the received symbols, and then use the most reliable basis elements to recover the source symbols with the inactivation decoding. Gaussian elimination is used to find the basis and to identify the most reliable basis elements. As a result, the BFA has polynomial time complexity. For random fountain codes, we are able to derive some theoretical bounds for the frame error rate (FER) of the BFA. Extensive simulations with Luby transform (LT) codes show that, the BFA has significantly lower FER than the belief propagation (BP) algorithm except for an extremely large amount of received symbols, and the FER of the BFA generally decreases as the average weight of basis elements increases.