Guruswami–Sudan list decoding algorithm


In coding theory, list decoding is an alternative to unique decoding of error-correcting codes in the presence of many errors. If a code has relative distance, then it is possible in principle to recover an encoded message when up to fraction of the codeword symbols are corrupted. But when error rate is greater than, this will not in general be possible. List decoding overcomes that issue by allowing the decoder to output a short list of messages that might have been encoded. List decoding can correct more than fraction of errors.
There are many polynomial-time algorithms for list decoding. In this article, we first present an algorithm for Reed–Solomon codes which corrects up to errors and is due to Madhu Sudan. Subsequently, we describe the improved Guruswami–Sudan list decoding algorithm, which can correct up to errors.
Here is a plot of the rate R and distance for different algorithms.
https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/81/Graph.jpg

Algorithm 1 (Sudan's list decoding algorithm)

Problem statement

Input : A field ; n distinct pairs of elements in ; and integers and.
Output: A list of all functions satisfying
is a polynomial in of degree at most
To understand Sudan's Algorithm better, one may want to first know another algorithm which can be considered as the earlier version or the fundamental version of the algorithms for list decoding RS codes - the Berlekamp–Welch algorithm.
Welch and Berlekamp initially came with an algorithm which can solve the problem in polynomial time with best threshold on to be.
The mechanism of Sudan's Algorithm is almost the same as the algorithm of Berlekamp–Welch Algorithm, except in the step 1, one wants to compute a bivariate polynomial of bounded degree. Sudan's list decoding algorithm for Reed–Solomon code which is an improvement on Berlekamp and Welch algorithm, can solve the problem with.This bound is better than the unique decoding bound for.

Algorithm

Definition 1
For weights, the – weighted degree of monomial is. The – weighted degree of a polynomial is the maximum, over the monomials with non-zero coefficients, of the – weighted degree of the monomial.
For example, has -degree 7
Algorithm:
Inputs: ; /* Parameters l,m to be set later. */
Step 1: Find a non-zero bivariate polynomial satisfying
Step 2. Factor Q into irreducible factors.
Step 3. Output all the polynomials such that is a factor of Q and for at least t values of

Analysis

One has to prove that the above algorithm runs in polynomial time and outputs the correct result. That can be done by proving following set of claims.
Claim 1:
If a function satisfying exists, then one can find it in polynomial time.
Proof:
Note that a bivariate polynomial of -weighted degree at most can be uniquely written as. Then one has to find the coefficients satisfying the constraints, for every. This is a linear set of equations in the unknowns. One can find a solution using Gaussian elimination in polynomial time.
Claim 2:
If then there exists a function satisfying
Proof:
To ensure a non zero solution exists, the number of coefficients in should be greater than the number of constraints. Assume that the maximum degree of in is m and the maximum degree of in is. Then the degree of will be at most. One has to see that the linear system is homogenous. The setting satisfies all linear constraints. However this does not satisfy, since the solution can be identically zero. To ensure that a non-zero solution exists, one has to make sure that number of unknowns in the linear system to be, so that one can have a non zero. Since this value is greater than n, there are more variables than constraints and therefore a non-zero solution exists.
Claim 3:
If is a function satisfying and is function satisfying and, then divides
Proof:
Consider a function. This is a polynomial in, and argue that it has degree at most. Consider any monomial of. Since has -weighted degree at most, one can say that. Thus the term is a polynomial in of degree at most . Thus has degree at most
Next argue that is identically zero. Since is zero whenever, one can say that is zero for strictly greater than points. Thus has more zeroes than its degree and hence is identically zero, implying
Finding optimal values for and.
Note that and
For a given value , one can compute the smallest for which the second condition holds
By interchanging the second condition one can get to be at most
Substituting this value into first condition one can get to be at least
Next minimize the above equation of unknown parameter. One can do that by taking derivative of the equation and equating that to zero
By doing that one will get,
Substituting back the value into and one will get

Algorithm 2 (Guruswami–Sudan list decoding algorithm)

Definition

Consider a Reed–Solomon code over the finite field with evaluation set and a positive integer, the Guruswami-Sudan List Decoder accepts a vector as input, and outputs a list of polynomials of degree which are in 1 to 1 correspondence with codewords.
The idea is to add more restrictions on the bi-variate polynomial which results in the increment of constraints along with the number of roots.

Multiplicity

A bi-variate polynomial has a zero of multiplicity at means that has no term of degree, where the x-degree of is defined as the maximum degree of any x term in

For example:
Let.
https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/76/Fig1.jpg
Hence, has a zero of multiplicity 1 at.
Let.
https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/76/Fig2.jpg
Hence, has a zero of multiplicity 1 at.
Let
https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/76/Fig3.jpg
Hence, has a zero of multiplicity 2 at.
Similarly, if
Then, has a zero of multiplicity 2 at.

General definition of multiplicity

has roots at if has a zero of multiplicity at when.

Algorithm

Let the transmitted codeword be, be the support set of the transmitted codeword & the received word be
The algorithm is as follows:
• Interpolation step
For a received vector, construct a non-zero bi-variate polynomial with weighted degree of at most such that has a zero of multiplicity at each of the points where
• Factorization step
Find all the factors of of the form and for at least values of
where & is a polynomial of degree
Recall that polynomials of degree are in 1 to 1 correspondence with codewords. Hence, this step outputs the list of codewords.

Analysis

Interpolation step

Lemma:
Interpolation step implies constraints on the coefficients of
Let
where and
Then, ........................
where
Proof of Equation 1:
Proof of Lemma:
The polynomial has a zero of multiplicity at if

Thus, number of selections can be made for and each selection implies constraints on the coefficients of

Factorization step

Proposition:
if is a factor of
Proof:
Since, is a factor of, can be represented as

where, is the quotient obtained when is divided by
is the remainder
Now, if is replaced by,
, only if
Theorem:
If, then is a factor of
Proof:
...........................From Equation 2

Given,
mod
Hence, mod
Thus, is a factor of.
As proved above,
where LHS is the upper bound on the number of coefficients of and RHS is the earlier proved Lemma.
Therefore,
Substitute,
Hence proved, that Guruswami–Sudan List Decoding Algorithm can list decode Reed-Solomon codes up to errors.