Generalized Codes on Graphs and Applications in Content Distribution

The contribution of this thesis is twofold. In the first part, we generalize and analyze two classes of error correcting codes: LDPC codes and product codes. We generalize graphical codes by considering checks being arbitrary codes instead of single parities. We analyze the performance of these codes under message passing decoding by using an extended density evolution over erasure channels. Further, we provide an optimal instance of these codes having rates arbitrarily close to the capacity of any binary symmetric channel where a belief propagation algorithm is used for decoding. The relation to other similar constructions is discussed as well. In a similar approach, we introduce irregular product codes, a class of codes where each codeword is represented by a matrix and the entries in each row (column) of the matrix come from a component row (column) code. Unlike product codes, we do not require that all component row and column codes be the same. This relaxation offers additional flexibility and features such as allowing some regions of the codeword to be more error-resilient, providing a more refined spectrum of rates for finite lengths, and improved performance for some of these rates. We study these codes over erasure channels and prove that for many rate distributions on component row codes, there is a matching rate distribution on component column codes such that an irregular product code based on MDS codes with those rate distributions on the component codes has an optimal asymptotic rate. In the second part we consider two novel applications of Raptor codes. Raptor codes are a class of fountain codes which present excellent performance, flexibility, and low decoding complexity over erasure channels. The data synchronization in an IP multicasting scenario involves reflecting changes from a content on a server to multiple clients efficiently. All these clients possess a slightly different copy of the content on the server. Existing methods are a direct extension of the point-to-point setting which is not necessarily optimal in terms of the total bandwidth usage in the multicast setting. We review this problem and its related information theoretic bounds. We propose and analyze an algorithm using Raptor codes which takes advantage of the multicast protocol to address this problem effectively. This method is close to optimal in terms of the amount of required transmitted information and the network usage. In another application, we show how to make satellite Digital Video Broadcasting resilient to signal jamming by distributing programs over multiple satellite links. However, leasing extra bandwidth on satellite links incurs supplementary cost to the broadcasting operator. In our solution we employ Raptor codes to make the bandwidth usage efficient and adaptive according to the link conditions. The proposed solution is fully compatible with the existing standards and infrastructure.