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Codes and Iterative Decoding on Algebraic Expander Graphs
 INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY AND ITS APPLICATIONS HONOLULU, HAWAII, U.S.A., NOVEMBER 58, 2000
, 2000
"... The notion of graph expansion was introduced as a tool in coding theory by Sipset and Spielman, who used it to bound the minimum distance of a class of lowdensity codes, as well as the performance of various iterative decoding algorithms for these codes. In spite of its usefulness in establishing t ..."
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The notion of graph expansion was introduced as a tool in coding theory by Sipset and Spielman, who used it to bound the minimum distance of a class of lowdensity codes, as well as the performance of various iterative decoding algorithms for these codes. In spite of its usefulness in establishing theoretical bounds on iterative decoding, graph expansion has not been widely used to design codes. Instead, random graphs are the primary means used to obtain graphs for codes, raising the question of whether comparable performance can be achieved using explicit constructions. In this paper we investigate the use of explicit algebraic expander graphs and algebraic subcodes, and show that the resulting coding schemes achieve excellent performance, competitive with standard lowdensity paritycheck codes over a wide range of block lengths. Since the code constructions are based on graphs of groups, the Fourier transform can be used to obtain fast encoding algorithms for these codes.
Double Coset Decompositions And Computational Harmonic Analysis On Groups
 JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
, 2000
"... In this paper we introduce new techniques for the efficient computation of a Fourier transform on a finite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorithms that generalize the CooleyTukey FFT to arbitrary finite groups. We appl ..."
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In this paper we introduce new techniques for the efficient computation of a Fourier transform on a finite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorithms that generalize the CooleyTukey FFT to arbitrary finite groups. We apply our general results to special linear groups and low rank symmetric groups, and obtain new efficient algorithms for harmonic analysis on these classes of groups, as well as the twosphere.
Probabilistic Decoding of LowDensity Cayley Codes
"... We report on some investigations into the behavior of a class of lowdensity codes constructed using algebraic techniques. Recent work shows expansion to be an essential property of the graphs underlying the lowdensity paritycheck codes first introduced by Gallager. In addition, it has recently be ..."
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We report on some investigations into the behavior of a class of lowdensity codes constructed using algebraic techniques. Recent work shows expansion to be an essential property of the graphs underlying the lowdensity paritycheck codes first introduced by Gallager. In addition, it has recently been shown that certain spectral techniques similar to those based on Fourier analysis for classical cyclic codes can be applied to codes constructed from Cayley graphs. This motivates us to compare the behavior of algebraically constructed expanders and randomly generated bipartite graphs using a probabilistic decoding algorithm. Preliminary results indicate that the performance of the explicit, algebraic expanders is comparable to that of random graphs in the case where each variable is associated with only two parity checks, while such codes are inferior to randomly generated codes with three or more constraints for each variable.
Decimationinfrequency Fast Fourier Transforms for the Symmetric Group
, 2005
"... In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a pathalgebraic approach to the representation theory of finite groups developed b ..."
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In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a pathalgebraic approach to the representation theory of finite groups developed by Ram (26). Applying this framework to the symmetric group, Sn, yields a class of fast Fourier transforms that we conjecture to run in O(n² n!) time. We also discuss several future directions for this research.
Spectral Graph Theory Lecture 12 Expander Codes
, 2009
"... In this lecture, I will show how Zemor [Zem01] used expander graphs to construct asymptotically good errorcorrecting codes and decode them efficiently. 12.2 Bipartite Expander Graphs Our construction of errorcorrecting codes will exploit bipartite expander graphs (as these give a much cleaner cons ..."
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In this lecture, I will show how Zemor [Zem01] used expander graphs to construct asymptotically good errorcorrecting codes and decode them efficiently. 12.2 Bipartite Expander Graphs Our construction of errorcorrecting codes will exploit bipartite expander graphs (as these give a much cleaner construction than the general case). Let’s begin by examining what a bipartite expander graph should look like. It’s vertex set will have two parts, U and V, each having n vertices. Every vertex will have degree d, and every edge will go from a vertex in U to a vertex in V. In the same way that we view ordinary expanders as approximations of complete graphs, we will view bipartite expanders as approximations of complete bipartite graphs 1. That is, if we let Kn,n denote the complete bipartite graph, then we want a dregular bipartite graph G such that (1 − ǫ) d n Kn,n � G � (1 + ǫ) d n Kn,n. As the eigenvalues of the Laplacian of d n Kn,n are 0 and 2d with multiplicity 1 each, and d otherwise, this means that we want a dregular graph G whose Laplacian spectrum satisfies λ1 = 0, λ2n = 2d, and λi − d  ≤ ǫd,for all 1 < i < 2n. We can obtain such a graph by taking the doublecover of an ordinary expander graph. Definition 12.2.1. Let G = (V,E) be a graph. The doublecover of G is the graph with vertex set V × {0,1} and edges ((u,0),(v,1)) , for (u,v) ∈ E. It is easy to determine the eigenvalues of the doublecover of a graph. Proposition 12.2.2. Let H be the doublecover of G. Then, for every eigenvalue λi of the Laplacian of G, H has a pair of eigenvalues, λi and 2d − λi. 1 The complete bipartite graph contains all edges between U and V