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In this paper, we consider the decoding of fountain codes where the receivedsymbols may have errors. It is motivated by the application of fountain codesin DNA-based data storage systems where the inner code decoding, whichgenerally has undetectable errors, is performed before the outer fountain codedecoding. We propose a novel and efficient decoding algorithm, namelybasis-finding algorithm (BFA), followed by three implementations. The key ideaof the BFA is to find a basis of the received symbols, and then use the mostreliable basis elements to recover the source symbols with the inactivationdecoding. Gaussian elimination is used to find the basis and to identify themost reliable basis elements. As a result, the BFA has polynomial timecomplexity. For random fountain codes, we are able to derive some theoreticalbounds for the frame error rate (FER) of the BFA. Extensive simulations withLuby transform (LT) codes show that, the BFA has significantly lower FER thanthe belief propagation (BP) algorithm except for an extremely large amount ofreceived symbols, and the FER of the BFA generally decreases as the averageweight of basis elements increases.

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