Analysis of the Second Moment of the LT Decoder

We analyze the second moment of the ripple size during the LT decoding process and prove that the standard deviation of the ripple size for an LT-code with length $k$ is of the order of $\sqrt k.$ Together with a result by Karp et. al stating that the expectation of the ripple size is of the order of $k$ [3], this gives bounds on the error probability of the LT decoder. We also give an analytic expression for the variance of the ripple size up to terms of constant order, and refine the expression in [3] for the expectation of the ripple size up to terms of the order of $1/k$, thus providing a first step towards an analytic finite-length analysis of LT decoding.Comment: 5 pages, 1 figure; submitted to ISIT 200

Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties

We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. The statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size, and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant

Asymmetric information embedding

In the Information Embedding Problem one is given a piece of data which can be altered only conditionally, for example only at certain places. One is then asked to embed an arbitrary message into the data by only applying admissible changes to the data. These changes lead to a distortion which is to be kept low. In this short note, we introduce an "asymmetric” version of information embedding in which the file is regarded as a string over a finite alphabet, and admissible changes on the alphabet elements are modeled by a directed graph. We introduce embedding techniques based on list-decoding algorithms for algebraic-geometric codes, and analyze their performanc

Representation theory for high-rate multiple-antenna code design

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels

Asymmetric information embedding

In the Information Embedding Problem one is given a piece of data which can be altered only conditionally, for example only at certain places. One is then asked to embed an arbitrary message into the data by only applying admissible changes to the data. These changes lead to a distortion which is to be kept low. In this short note, we introduce an ?asymmetric? version of information embedding in which the file is regarded as a string over a finite alphabet, and admissible changes on the alphabet elements are modeled by a directed graph. We introduce embedding techniques based on list-decoding algorithms for algebraic-geometric codes, and analyze their performance

Coding schemes for line networks

Abstract — We consider a simple network, where a source and destination node are connected with a line of erasure channels. It is well known that in order to achieve the min-cut capacity, the intermediate nodes are required to process the information. We propose coding schemes for this setting, and discuss each scheme in terms of complexity, delay, achievable rate, memory requirement, and adaptability to unknown channel parameters. We also briefly discuss how these schemes can be extended to more general networks. I