Capstone Project: LT Codes

Erasure codes are a type of encoding of data, generally for transmission over a lossy medium, that survive deletions (erasures) of parts of the message.1 They are especially useful for transmission of data across a medium or network that can drop packets of data, when it is impractical for the receiver to be in constant communication with the sender. Given a file with K blocks, the sender generates B > K encoded blocks (B/K is called the rate of the code). The code is designed so that after receiving any set of blocks of size at least K ′, for some K ′ slightly larger than K, the original data can be decoded with high probability. For scenarios like broadcast, or for networks with very long one-way delays (think the Mars rover sending an image to Earth), this is much more practical than the receiver acknowledging every block, as done in the Transmission Control Protocol (TCP), which is used on the web and by many other internet applications. Reed-Solomon codes are a type of erasure code, but are not very practical for many applications, in particular because you have to set the rate prior to encoding and transmitting. The problem is that the rate might need to change depending on the quality of the channel at the receiver(s)! There exist codes, however, that are rateless, in that a practically infinite number of coded blocks can be generated from a fixed set of source blocks, and the receiver can still can decode the original set of blocks with little overhead. These codes are also called fountain codes, in an analogy to a constant stream of water from a fountain; any set of drops from the fountain will serve the purpose of filling the receiver’s bucket. In this project you will implement LT (Luby transform) Codes, which were the first practical rateless erasure codes, and were invented in 1998 by Michael Luby and colleagues [1]. We will base our description on chapter 50 of the book by MacKay [2], which is freely available for download. We recommend that you read that chapter before beginning your project. In particular, you will need to understand the encoding and decoding algorithms described in Sec. 50.1-50.2. We will use the distributions described in Sec. 50.3, but you do not need to understand their justification. LT Codes form the basis of the current state of the art in rateless codes, called Raptor Codes. Raptor codes are faster than LT Codes, and generally require fewer blocks to decode, with K ′ very close to K. They are used in several communication standards such as in broadcast of video to mobile devices. Their implementation, however, is more involved.