We present a new class of asymptotically good, linear error-correcting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel decoding algorithms with a linear number of processors, and are simple to understand. We present both randomized and explicit constructions for some of these codes. Experimental results demonstrate the extremely good performance of the randomly chosen codes. 1. Introduction We present a new class of error correcting codes derived from expander graphs. These codes have the advantage that they can be decoded very efficiently. That makes them particularly suitable for devices which must decode cheaply, such as compact disk players and remote satellite receivers. We hope that the connection we draw between expander graphs and error correcting codes will stimulate research in both fields. 1.1. Error correcting codes An error correcting code is a mapping from messages to codewords such that the mappi...

Let G = (V; E) be an undirected weighted graph with jV j = n and jEj = m. Let k 1 be an integer. We show that G = (V; E) can be preprocessed in O(kmn ) expected time, constructing a data structure of size O(kn ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k \Gamma 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k \Gamma 1. We show that a 1963 girth conjecture of Erdos, implies ) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal.

Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 De-randomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 De-randomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of every size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as “entropy wave ” propagators — they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves. A variant of this product can be applied to extractors, giving the first explicit extractors whose seed length depends (poly)logarithmically on only the entropy deficiency of the source (rather than its length) and that extract almost all the entropy of high min-entropy sources. These high min-entropy extractors have several interesting applications, including the first constant-degree explicit expanders which beat the “eigenvalue bound.” Keywords: expander graphs, extractors, dispersers, samplers, graph products

A new technique, based on the pseudo-random properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping the code coordinates. For any fixed (small) rate, and for sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF (2)) as well. Although these concatenated codes lie below Zyablov bound, they are still superior to previously-known explicit constructions in the zero-rate neighborhood.

We present a new class of asymptotically good, linear error-correcting codes. These codes can be both encoded and decoded in linear time. They can also be encoded by logarithmic-depth circuits of linear size and decoded by logarithmic depth circuits of size 0 (n log n). We present both randomized and explicit constructions of these codes.

A d-regular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random d-regular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 +ɛwith probability 1 − O(n−τ), where τ = ⌈ � √ d − 1+1 � /2⌉−1. We also show that this probability is at most 1 − c/nτ ′, for a constant c and a τ ′ that is either τ or τ +1 (“more often ” τ than τ + 1). We prove related theorems for other models of random graphs, including models with d odd. These theorems resolve the conjecture of Alon, that says that for any ɛ>0andd, the second largest eigenvalue of “most ” random dregular graphs are at most 2 √ d − 1+ɛ (Alon did not specify precisely what “most ” should mean or what model of random graph one should take). 1

Berman and Schnitger [10] gave a randomized reduction from approximating MAXSNP problems [24] within constant factors arbitrarily close to 1 to approximating clique within a factor of n^ε (for some ε). This reduction was further studied by Blum [11], who gave it the name randomized graph products. We show that this reduction can be made deterministic (derandomized), using random walks on expander graphs [1]. The main technical contribution of this paper is in lower bounding the probability that all steps of a random walk stay within a specified set of vertices of a graph. (Previous work was mainly concerned with upper bounding this probability.) This lower bound extends also to the case that different sets of vertices are specified for different time steps of the walk.

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma, Umans, and Zuckerman (STOC ‘01) required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable errorcorrecting codes of Parvaresh and Vardy (FOCS ‘05). Our expanders can be interpreted as near-optimal “randomness condensers, ” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. (STOC ‘03) and improving upon it when the error parameter is small (e.g. 1/poly(n)).

The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of k-regular Ramanujan graphs. We improve the lower bound on the expansion of Ramanujan graphs to approximately k=2. Moreover, we construct a family of k-regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k=2. This shows that k=2 is the best bound one can obtain using the second eigenvalue method. We also show an upper bound of roughly 1 + p k \Gamma 1 on the average degree of linear-sized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2 p k \Gamma 1. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (resp. extrovert graphs) of smaller size (resp. degree) th...