Graph-based Codes and Generalized Product Constructions

Fountain codes are a class of rateless codes well-suited for communication over erasure channels with unknown erasure parameter. We analyze the decoding of LT codes, the first instance of practical, universal Fountain codes. A set of interest during LT decoding is the set of output symbols of reduced degree one (the “ripple”). The evolution of the ripple size throughout the decoding process is crucial to successful decoding. We derive an expression for the variance of the ripple size, thus obtaining bounds on the error probability of the decoder. This provides a first step toward finite-length analysis of LT codes, and ultimately toward the design of provably good Fountain codes. In a related work, we analyze a generalization of LT codes where output symbols consist of r parities generated from a subset of input symbols via an MDS code. We study the modified decoding algorithm and derive a Soliton-like distribution that allows recovering of the input symbols with high probability with asymptotically vanishing overhead. We study two generalizations of product codes. First, we introduce and analyze irregular product codes, a generalization where rows and columns of codeword matrices are not restricted to a single row and column code but can come from a distribution of component codes of varying rates. We characterize the rate of these codes and give families of irregular product codes based on MDS component codes that achieve rate 1 − ε on erasure channels of parameter ε (at the cost of a growing field size). We also exhibit finite-length irregular product codes that outperform all regular product codes of similar rate on erasure channels. Second, we study staircase codes, a product-like construction introduced in [1] that combines a short component code in two dimensions in a “continuous” fashion. Motivated by the improvement that these codes present over their component code, we seek to push this improvement further by going to higher dimensions still. We abstract out the properties of staircase codes that allow them to be generalized to higher dimensions and design a three-dimensional staircase code. In the field of network coding, we propose a relaxed measure of security over untrusted networks, where intermediary nodes are allowed to learn only as many linear combinations of the source messages as they need to send. We study the problem over some classes of combination networks and give necessary and sufficient conditions for achieving this notion of security over such networks.