Raptor code

In computer science, raptor codes are one of the first known classes of fountain codes with linear time encoding and decoding. They were invented by Amin Shokrollahi in 2000/2001 and were first published in 2004 as an extended abstract.

Raptor codes encode a given message consisting of a number of symbols, "k", into a potentially limitless sequence of encoding symbols such that knowledge of any "k" or more encoding symbols allows the message to be recovered with some non-zero probability. The probability that the message can be recovered increases with the number of symbols received above "k" becoming very close to 1, once the number of received symbols is only very slightly larger than "k". A symbol can be any size, from a single bit to hundreds or thousands of bytes.

Raptor codes may be systematic or non-systematic. In the systematic case, the symbols of the original message are included within the set of encoding symbols. An example of a systematic raptor code is the code defined by the 3rd Generation Partnership Project for use in mobile cellular wireless broadcast and multicast and also used by DVB-H standards for IP datacast to handheld devices (see external links). Online codes are an example of a non-systematic raptor code.

Overview

Raptor codes are formed by the concatenation of two codes.

A fixed rate erasure code, usually with a fairly high rate, is applied as a 'pre-code' or 'outer code'. This pre-code may itself be a concatenation of multiple codes, for example in the code standardized by 3GPP a high density parity check code derived from the binary Gray sequence is concatenated with a simple regular low density parity check code. Another possibility would be a concatenation of a Hamming code with a low density parity check code.

The inner code takes the result of the pre-coding operation and generates a sequence of encoding symbols. The inner code is a form of LT code. Each encoding symbol is the XOR of a randomly chosen set of symbols from the pre-code output. The number of symbols which are XOR'ed together to form an output symbol is chosen randomly for each output symbol according to a specific probability distribution.

This distribution, as well as the mechanism for generating random numbers for sampling this distribution and for choosing the symbols to be XOR'ed, must be known to both sender and receiver. In one approach, each symbol is accompanied with an identifier which can be used as a seed to a pseudo-random number generator to generate this information, with the same process being followed by both sender and receiver.

In the case of non-systematic raptor codes, the source data to be encoded is used as the input to the pre-coding stage.

In the case of systematic raptor codes, the input to the pre-coding stage is obtained by first applying the inverse of the encoding operation that generates the first "k" output symbols to the source data. Thus, applying the normal encoding operation to the resulting symbols causes the original source symbols to be regenerated as the first "k" output symbols of the code. It is necessary to ensure that the randomprocesses which generate the first "k" output symbols generate an operation which is invertible.

Decoding

Two approaches are possible for decoding raptor codes. In a concatenated approach, the inner code is decoded first, using a belief propagation algorithm, as used for the LT codes. Decoding succeeds if this operation recovers a sufficient number of symbols, such that the outer code can recover the remaining symbols using the decoding algorithm appropriate for that code.

In a combined approach, the relationships between symbols defined by both the inner and outer codes are considered as a single combined set of simultaneous equations which can be solved by the usual means, typically by Gaussian elimination.

Computational complexity

Raptor codes require "O(1)" time to generate an encoding symbol and "O(k)" time to decode a message of length "k".

See also

* Erasure code
* LT code
* Graphical code
* Michael Luby

References

* Amin Shokrollahi, "Raptor Codes," IEEE Transactions on Information Theory, vol. 52, pp. 2551-2567, 2006. [http://ieeexplore.ieee.org/iel5/18/34354/01638543.pdf?isnumber=34354&prod=JNL&arnumber=1638543&arSt=2551&ared=2567&arAuthor=Shokrollahi%2C+A. PDF]
* [http://www.3gpp.org 3GPP] The 3rd Generation Partnership Project
* [http://www.dvb.org DVB] Digital Video Broadcasting
* [http://www.3gpp.org/ftp/Specs/html-info/26346.htm 3GPP TS26.346] 3GPP Technical Specification for Multimedia Broadcast/Multicast Service: Protocols and Codecs.
* [http://www.dvb-h-online.org/technology.htm DVB-H IP Datacasting specifications]