# Snell envelope

In coding theory, list decoding is an alternative to unique decoding of error-correcting codes for large error rates. Using unique decoder one can correct up to fraction of errors. But when error rate is greater than , unique decoder will not able to output the correct result. List decoding overcomes that issue. List decoding can correct more than fraction of errors.

There are many efficient algorithms that can perform List decoding. list decoding algorithm for Reed–Solomon (RS) codes by Sudan which can correct up to errors is give first. Later on more efficient Guruswami–Sudan list decoding algorithm, which can correct up to errors is discussed.

Here is the plot between 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 from ; and integers and .

**Output:** A list of all functions satisfying

is a polynomial in of degree at most d with | { } | -- (1)

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 (weighted degree)**

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.

E.g. : is a monomial in variables with a coefficient of 3.

**Algorithm:**

Inputs: ; {} /* Parameters l,m to be set later. */

Step 1: Find any function satisfying
has (1,*d*)-weighted degree at most , (2)
for every n,
is not identically zero.

Step 2. Factor the polynomial Q into irreducible factors.

Step 3. Output all the polynomials such that is a factor of Q and for at least t values of n

### Analysis

One have 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 (2) exists, then one can find it in polynomial time.

**Proof:**

Note that a bivariate polynomial of degree at most can be represented as follows: Let . Then one have 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 (2)

**Proof:**

To ensure a non zero solution exists, the number of variables in should be greater than the number of constraints. Assume that maximum degree of in be m and maximum degree of in be l. Then the degree of will be atmost . One have to see that the linear system is homogenous. The setting satisfies all linear constraints. However this does not satisfy (2), since the solution can be identically zero. To ensure that non-zero solutions exists, One have 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 (2) and is function satisfying (1) 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

https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/76/Fig1.jpg

Hence, has a zero of multiplicity 1 at (0,0).

https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/76/Fig2.jpg

Hence, has a zero of multiplicity 1 at (0,0).

https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/76/Fig3.jpg

Hence, has a zero of multiplicity 2 at (0,0).

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

Then, ........................(Equation 1)

**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:**

**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:**

**Proof:**

...........................From Equation 2

As proved above,

where LHS is the upper bound on the number of coefficients of and RHS is the earlier proved Lemma.

Hence proved, that Guruswami–Sudan List Decoding Algorithm can list decode Reed-Solomon(RS) codes up to errors.

## References

- http://www.cse.buffalo.edu/~atri/courses/coding-theory/
- http://www.cs.cmu.edu/~venkatg/pubs/papers/listdecoding-NOW.pdf
- http://www.mendeley.com/research/algebraic-softdecision-decoding-reedsolomon-codes/
- R. J. McEliece. The Guruswami-Sudan Decoding Algorithm for Reed-Solomon Codes.
- M Sudan. Decoding of Reed Solomon codes beyond the error-correction bound.