We know that a Reed Solomon code (RS code),
, has a distance
and a list decoding radius
. Hence the list decoding rate, R, of the code is
, where
is the number of fraction errors. Now the next question is can we arithmetically do better than this in polynomial time? For seven years after the Guruswami-Sudan, there was no progress till the break through work of Parvaresh and Vardy. The Parvaresh-Vardy (PV) codes are based on RS codes.
The list decoding algorithm is based on two key ideas. First is the transition from bi-variate polynomial interpolation to multivariate interpolation decoding. The second key idea is to take different approach than that is taken with RS codes as a number of prior attempts to overcome the
rate barrier has already proved unsuccessful. Hence rather than devising a better list-decoder for RS codes, new codes were constructed.
Standard RS encoder view a message as a polynomial
over a field
, and produce the corresponding codeword by evaluating
at
distinct elements of
. In case of PV codes, given
, first related polynomials
are computed and then the corresponding codeword is produced by evaluating all these polynomials. Correlation between
and
is of form
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. Here
is an arbitrary irreducible (over
) polynomial of degree k, and
is an arbitrary (but sufficient large) integer. Correlation between
and
when
provides information that is exploited to break the
fraction of error barrier for adversarial errors.
Input for the PV list decoding algorithm will be
and
. Here
is called the agreement parameter. The output of algorithm will be all degree
polynomials
such that the PV codeword corresponding to
agrees with the received word in at least
places. The algorithm consists of the following steps,
- Find
such that
for all
.
.
- While
, put
.
- Put
, put
and output all roots of
.
Here note that
and if
and
then
.
In case of PV codes the message corresponds to an element of
, i.e.
symbols from the alphabet
. Hence rate is
. As we can list decode from
agreement, hence we could recover from
fraction of errors. On the other hand RS codes achieved only a rate of
.
Some improvements that can be made to PV codes include,
- We can insist that
have multiple roots at
. This would eliminate the leading constant factor of 3 in
, and would improve rate to
.
- Additionally we can use correlated polynomials to extract additional performance from PV codes. Let
be the message that we want to transmit. For
we put
. This results in following encoding,
.
Now although we pay extra running time in
while decoding, but its still remains a polynomial time algorithm for any fixed m and yield recovery from
fraction of errors. Asymptotically, for large
, this approaches (letting
now)
but doesn’t really do much better for any fixed R. Also since alphabet becomes
-tuples of
, rate can not possibly increase
.
Application in Guruswami-Umans-Vadhan Expander Construction
[edit]
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.
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.
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.
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
.
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:
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
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,
since
.
Then
which is an element of the extended field
where
is an irreducible polynomial of degree ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
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.
- Farzad Parvaresh and Alexander Vardy. 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.
- Atri Rudra. Error Correcting Codes: Combinatorics, Algorithms and Applications Lecture 41
- Madhu Sudan. Essential coding theory Lecture 15 and Lecture 16
- Unbalanced Expanders and Randomness Extractors from Parvaresh–Vardy Codes - GUV paper.
- Expander graphs.
- Expander graphs and vertex expansion.
- Bipartite graph.
- Digraph.
- Polynomials and their properties.
- This article incorporates text from a comparable page at the website of SUNY Buffalo