Overview

Expanders are highly connected yet sparse graphs. They have a wide variety of applications in theoretical computer science, in designing algorithms, to construct hash functions in cryptography, error correcting codes, extractors, pseudorandom generators, sorting networks and robust computer networks.
The construction of expanders of Guruswami-Umans-Vadhan is based on the list decodable codes of Parvaresh and Vardy.

Introduction

Let us review the basics of list decodable codes. We take C as the code which is a mapping encoding messages of bit length to symbols over the alphabet Rate of such a code will be . We call as list decodable if for every, the set LIST is of atmost K size. With list decodable codes, we wish to optimize the tradeoff between the agreement and the rate which do not depend on message length M.
Sudan showed that such a property can be achieved by Reed Solomon Codes in polynomial time. This tradeoff was then improved by Guruswami and Sudan and recently by Parvaresh and Vardy who improved the tradeoff by using a variant of Reed Solomon codes.

GUV Constructor

The construction of Guruswami-Umans-Vadhan Expander is based on Parvaresh Vardy codes.We know that a typical Parvaresh Vardy codeword has several related degree polynomials evaluated at all points in the field and where is a prime power over which the field is defined. All such evaluations are packaged into larger alphabet symbol. This extra redundancy enables a better list decoding algorithm than Reed Solomon ones.
Elements of are chosen such that for and integer parameter.

We need to show that for a given set of size , the set LIST is small.

Expander Graphs

Lets start with some definitions : For a bipartite graph and a set , define .
Also, a digraph is a vertex expander if for all sets of at most vertices, the neighborhood is of size atleast where neighborhood s.t. . Details can be found out in the paper Expander graphs and vertex expansion.
This proves the following lemma:

Lemma- A graph is a expander if and only if for every set of size at most is of size at most .

Construction

Fix the field and let be an irreducible polynomial of degree over the field . Elements of are univariate polynomials over with degree at most . , integer parameter is fixed.
The expander is bipartite graph defined as:
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The bipartite graph has message polynomials on the left and the neighbor of is the symbol of Parvaresh-Vardy encoding of . This follows a theorem which can formally be stated as:

Theorem 1: The graph is a expander for and .

Proof: Let us take any integer , where and let . By the lemma defined above, if we take a such that is of at most size, then we need to show that .

Parvaresh-Vardy codes view degree polynomials as elements of field where is an irreducible polynomial of degree . We need that will have non zero coefficients on monomials of the form for and , where and is the base- representation of . If we impose a homogeneous linear constraint on coefficients of , then we require that for every . Since number of constraints is less than the number of unknowns, the linear system thus made has a solution that is not 0. If has the smallest possible degree in variable , then

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for univariate polynomials , at least one of will not be divisible by . If every is divisible by then will have smaller degree in and would still vanish on (since is irreducible and therefore has no roots in ).

Let us take to be any polynomial. Then by our ,
.
This means, the univariate polynomial has zeroes. Since has at most degree , then it is . Refer Polynomials and properties for proof. So,

Recall that, we have, . Thus,
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Then which is an element of the extended field where is an irreducible polynomial of degree is the root of univariate polynomial over defined by

From equation , the above equation is same as:

Since this is true for all , has at least roots in field . Some 's is not divisible by , is a non zero polynomial. Thus, is bounded by the degree of , which is at most .

By proper instantiation of parameters in Theorem 1, we lead to following results:
Theorem 2: For all positive integers , , all , and all for , there is an explicit expander with degree and . Moreover, and are powers of .

Theorem 3: For all positive integers , , and all , there is an explicit expander with degree and . Again, and are powers of ..

The proofs of the above two theorems can be found from GUV paper.

References

1. Unbalanced Expanders and Randomness Extractors from Parvaresh–Vardy Codes - GUV paper.

3. Farzad Parvaresh and Alexander Vardy. [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?
arnumber=1530722&contentType=Conference+Publications Correcting errors beyond the Guruswami-Sudan radius in polynomial time] In
Proceedings of the 43nd Annual Symposium on Foundations of Computer Science (FOCS), pages 285-294, 2005.

4. Atri Rudra. Error Correcting Codes: Combinatorics, Algorithms and Applications Lecture 41

5. Madhu Sudan. Essential coding theory Lecture 15 and Lecture 16

8. digraph.

9. Polynomials and their properties.