We study two families of error-correcting codes defined in terms of very sparse matrices "MN" (MacKay-Neal (1995)) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties The decoding of both codes can be tackled with a practical sum-product algorithm We prove that these codes are "very good", in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit This result holds not only for the binary-symmetric channel but also for any channel with symmetric stationary ergodic noise We give experimental results for binary-symmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes

/pdf/good-error-correcting-codes-based-on-very-sparse-matrices-3kdz0lhhm4.pdf

Good error-correcting codes based on very sparse matrices

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

We present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of the given ensemble will achieve an arbitrarily small target probability of error with a probability that approaches one exponentially fast in the length of the code. (By concatenating with an appropriate outer code one can achieve a probability of error that approaches zero exponentially fast in the length of the code with arbitrarily small loss in rate.) Conversely, transmitting at rates above this capacity the probability of error is bounded away from zero by a strictly positive constant which is independent of the length of the code and of the number of iterations performed. Our results are based on the observation that the concentration of the performance of the decoder around its average performance, as observed by Luby et al. in the case of a binary-symmetric channel and a binary message-passing algorithm, is a general phenomenon. For the particularly important case of belief-propagation decoders, we provide an effective algorithm to determine the corresponding capacity to any desired degree of accuracy. The ideas presented in this paper are broadly applicable and extensions of the general method to low-density parity-check codes over larger alphabets, turbo codes, and other concatenated coding schemes are outlined.

/pdf/the-capacity-of-low-density-parity-check-codes-under-message-4plluv9kib.pdf

The capacity of low-density parity-check codes under message-passing decoding

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even

/pdf/expander-graphs-and-their-applications-wbsb8gxfw2.pdf

Expander graphs and their applications

/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1. Algebraic Groups Over Arbitrary Fields.- 2. Algebraic Groups Over Local Fields.- 3. Arithmetic Groups.- 4. Measure Theory and Ergodic Theory.- 5. Unitary Representations and Amenable Groups.- II. Density and Ergodicity Theorems.- 1. Iterations of Linear Transformations.- 2. Density Theorems for Subgroups with Property (S)I.- 3. The Generalized Mautner Lemma and the Lebesgue Spectrum.- 4. Density Theorems for Subgroups with Property (S)II.- 5. Non-Discrete Closed Subgroups of Finite Covolume.- 6. Density of Projections and the Strong Approximation Theorem.- 7. Ergodicity of Actions on Quotient Spaces.- III. Property (T).- 1. Representations Which Are Isolated from the Trivial One-Dimensional Representation.- 2. Property (T) and Some of Its Consequences. Relationship Between Property (T) for Groups and for Their Subgroups.- 3. Property (T) and Decompositions of Groups into Amalgams.- 4. Property (R).- 5. Semisimple Groups with Property (T).- 6. Relationship Between the Structure of Closed Subgroups and Property (T) of Normal Subgroups.- IV. Factor Groups of Discrete Subgroups.- 1. b-metrics, Vitali's Covering Theorem and the Density Point Theorem.- 2. Invariant Algebras of Measurable Sets.- 3. Amenable Factor Groups of Lattices Lying in Direct Products.- 4. Finiteness of Factor Groups of Discrete Subgroups.- V. Characteristic Maps.- 1. Auxiliary Assertions.- 2. The Multiplicative Ergodic Theorem.- 3. Definition and Fundamental Properties of Characteristic Maps.- 4. Effective Pairs.- 5. Essential Pairs.- VI. Discrete Subgroups and Boundary Theory.- 1. Proximal G-Spaces and Boundaries.- 2. ?-Boundaries.- 3. Projective G-Spaces.- 4. Equivariant Measurable Maps to Algebraic Varieties.- VII. Rigidity.- 1. Auxiliary Assertions.- 2. Cocycles on G-Spaces.- 3. Finite-Dimensional Invariant Subspaces.- 4. Equivariant Measurable Maps and Continuous Extensions of Representations.- 5. Superrigidity (Continuous Extensions of Homomorphisms of Discrete Subgroups to Algebraic Groups Over Local Fields).- 6. Homomorphisms of Discrete Subgroups to Algebraic Groups Over Arbitrary Fields.- 7. Strong Rigidity (Continuous Extensions of Isomorphisms of Discrete Subgroups).- 8. Rigidity of Ergodic Actions of Semisimple Groups.- VIII. Normal Subgroups and "Abstract" Homomorphisms of Semisimple Algebraic Groups Over Global Fields.- 1. Some Properties of Fundamental Domains for S-Arithmetic Subgroups.- 2. Finiteness of Factor Groups of S-Arithmetic Subgroups.- 3. Homomorphisms of S-Arithmetic Subgroups to Algebraic Groups.- IX. Arithmeticity.- 1. Statement of the Arithmeticity Theorems.- 2. Proof of the Arithmeticity Theorems.- 3. Finite Generation of Lattices.- 4. Consequences of the Arithmeticity Theorems I.- 5. Consequences of the Arithmeticity Theorems II.- 6. Arithmeticity, Volume of Quotient Spaces, Finiteness of Factor Groups, and Superrigidity of Lattices in Semisimple Lie Groups.- 7. Applications to the Theory of Symmetric Spaces and Theory of Complex Manifolds.- Appendices.- A. Proof of the Multiplicative Ergodic Theorem.- B. Free Discrete Subgroups of Linear Groups.- C. Examples of Non-Arithmetic Lattices.- Historical and Bibliographical Notes.- References.

Discrete subgroups of semisimple Lie groups

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices.

https://people.brandeis.edu/~kleinboc/Pub/flda.pdf

Flows on homogeneous spaces and Diophantine approximation on manifolds

We give an explicit construction of regular graphs of degree 2r withn vertices and girth ≧c logn/logr. We use Cayley graphs of factor groups of free subgroups of the modular group. An application to low density codes is given.

Explicit constructions of graphs without short cycles and low density codes

In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {A
 t
  | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f
 t
 of G under which #{t∈ℕ | f
 t
 
x∈A
 t
 } is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov.

https://people.brandeis.edu/~kleinboc/Pub/lld.pdf

Logarithm laws for flows on homogeneous spaces

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.

Gregory Margulis

Papers

Discrete subgroups of semisimple Lie groups

Flows on homogeneous spaces and Diophantine approximation on manifolds

Explicit constructions of graphs without short cycles and low density codes

Logarithm laws for flows on homogeneous spaces

Flows on homogeneous spaces and Diophantine approximation on manifolds