Rateless erasure code

In coding theory, fountain codes (also known as rateless erasure codes) are a class of erasure codes with the property that a potentially limitless sequence of encoding symbols can be generated from a given set of source symbols such that the original source symbols can ideally be recovered from any subset of the encoding symbols of size equal to or only slightly larger than the number of source symbols. The term fountain or rateless refers to the fact that these codes do not exhibit a fixed code rate.

A fountain code is optimal if the original k source symbols can be recovered from any k encoding symbols. Fountain codes are known that have efficient encoding and decoding algorithms and that allow the recovery of the original k source symbols from any k’ of the encoding symbols with high probability, where k’ is just slightly larger than k.

LT codes were the first practical realization of fountain codes. Raptor codes and online codes were subsequently introduced, and achieve linear time encoding and decoding complexity through a pre-coding stage of the input symbols.

Fountain codes are flexibly applicable at a fixed code rate, or where a fixed code rate cannot be determined a priori, and where efficient encoding and decoding of large amounts of data is required.

One example is that of a data carousel, where some large file is continuously broadcast to a set of receivers. Using a fixed-rate erasure code, a receiver missing a source symbol (due to a transmission error) faces the coupon collector's problem: it must successfully receive an encoding symbol which it does not already have. This problem becomes much more apparent when using a traditional short-length erasure code, as the file must be split into several blocks, each being separately encoded: the receiver must now collect the required number of missing encoding symbols for each block. Using a fountain code, it suffices for a receiver to retrieve any subset of encoding symbols of size slightly larger than the set of source symbols. (In practice, the broadcast is typically scheduled for a fixed period of time by an operator based on characteristics of the network and receivers and desired delivery reliability, and thus the fountain code is used at a code rate that is determined dynamically at the time when the file is scheduled to be broadcast.)

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