The Berlekamp–Welch algorithm, also known as the Welch–Berlekamp algorithm, is named for Elwyn R. Berlekamp and Lloyd R. Welch. The algorithm efficiently corrects errors in BCH codes and Reed–Solomon codes (which are a subset of BCH codes). Unlike many other decoding algorithms, and in correspondence with the code-domain Berlekamp–Massey algorithm that uses syndrome decoding and the dual of the codes, the Berlekamp–Welch decoding algorithm provides a method for decoding Reed–Solomon codes using just the generator matrix and not syndromes.
History on decoding Reed–Solomon codes
- In 1960, Peterson developed an algorithm for decoding BCH codes. His algorithm solves the important second stage of the generalized BCH decoding procedure and is used to calculate the error locator polynomial coefficients that in turn provide the error locator polynomial. This is crucial to the decoding of BCH codes.
- In 1963, Gorenstein–Zierler saw that BCH codes and Reed–Solomon codes have a common generalization and that the decoding algorithm extends to more general situation.
- In 1968 / 69, Elwyn Berlekamp invented an algorithm for decoding BCH codes. James Massey recognized its application to linear feedback shift registers and simplified the algorithm. Massey termed the algorithm the LFSR Synthesis Algorithm (Berlekamp Iterative Algorithm) but it is now known as the Berlekamp–Massey algorithm.
- In 1975, Sugiyama et al. developed a decoder based on the extended Euclidean algorithm. Reed–Solomon_error_correction#Euclidean_decoder
- In 1986, The Welch–Berlekamp algorithm was developed to solve the decoding equation of Reed–Solomon codes, using a fast method to solve a certain polynomial equation. The Berlekamp–Welch algorithm has a running time complexity of . The following sections look at the Gemmel and Sudan’s exposition of the Berlekamp–Welch algorithm.
Error locator polynomial of Reed–Solomon codes
In the problem of decoding Reed–Solomon codes, the inputs are pair wise distinct evaluation points where with dimension and distance and a codeword Our goal is to describe an algorithm that can correct many errors in polynomial time. To do so we have to find such that and the number of indices for which is less than or equal to We can assume that there exists a polynomial such that
Note that the coefficients of are the encoded information. To solve this, we use an indicator for those indices where an error may have occurred. Thus we define an error locator polynomial, by:
Note that We can also claim that holds for all . This fact holds true because in the event of , both sides of the above equation vanish because .
However, since and are both unknown, the main task of the decoding algorithm would be to find . To do this we use a seemingly useless yet very powerful method and define another polynomial This is because the equations with we need to solve are quadratic in nature. Thus by defining a product of two variables that gives rise to a quadratic term as one unknown variable, we increase the number of unknowns but make the equations linear in nature. This method is called linearization and is a very powerful tool.
Thus having the properties:
This helps because if we now manage to find and , we can easily find using . The main purpose of the Berlekamp Welch algorithm is to find out using degree bounded polynomials and and the properties of and .
Computing is as hard as finding the end solution Once is computed, using erasure decoding for Reed–Solomon codes, we can easily recover . However, in a few cases, even the polynomial is as hard to find as . As an example, given and (such that for ), by checking positions where , we can find the error locations. Thus the algorithm works on the principle that while each of the polynomials and are hard to find individually; computing them together is much easier.
The Berlekamp–Welch decoder and algorithm
The Welch–Berlekamp decoder for Reed–Solomon codes consists of the Welch– Berlekamp algorithm augmented by some additional steps that prepare the received word for the algorithm and interpret the result of the algorithm.
The inputs given to the Berlekamp Welch decoder are the integers denoting Block Length the number of errors such that and the received word satisfying the condition that there exists at most one with with .
The output of the decoder is either the polynomial , or in some cases, a failure. This decoder functions in two steps as follows:
- This step is called the interpolation step in which the decoder computes a non zero polynomial of degree (This implies that the coefficient of must be 1) and another polynomial with These polynomials are created such that the condition holds for all In the case that polynomials satisfying the above condition cannot be computed, the output of the decoder would be a failure.
- If then a is defined which equals If then the decoder outputs If the above condition is not satisfied, i.e. if then a failure is returned by the decoder.
According to the algorithm, in the cases where it does not output a failure, it outputs a that is the correct and desired polynomial. To prove that, the algorithm always outputs the desired polynomial, we need to prove a few claims we have made while describing the algorithm. Let us go ahead and do so now.
- Claim 1. There exist a pair of polynomials, that satisfy Step 1 of the BW algorithm and
Let be the error-locating polynomial for :
Notice that has the following properties by definition:
Now define and note that:
We can now claim that from the first step of the BW algorithm holds. If then . For we have and therefore just as we claimed.
This above claim however just reiterates and proves the fact that there exists a pair of polynomials and such that It however does not necessarily guarantee the fact that the algorithm we discussed above would indeed output such a pair of polynomials. We therefore move on to look at another claim that helps establish this fact using the above claim and thereby proving the correctness of the algorithm.
- Claim 2. If are two distinct solutions that satisfy the first step of the Berlekamp Welch algorithm, then we have
First note that
Then we define:
Note that From step 1 of the Berlekamp Welch algorithm we also know that and Now for all we calculate:
Thus has roots, on the other hand
Therefore, is the zero polynomial which means that and are identical. Since are non-zero we can write: as per our initial claim.
Thus based on the above claims, we can safely state that the output of the Berlekamp Welch algorithm, when outputting the polynomial is correct.
We can now claim that the algorithm can be implemented such that it has a running time of . This can be proved as follows: In Step 1 of the algorithm, the polynomials and have and unknown values respectively and the constraints for all acts as a linear equation with these unknowns. We therefore get a system of linear equations in unknowns. Using our first claim, this system of equations has a solution since This can be solved in time, by say Gaussian elimination. Finally, we can note that Step 2 of the algorithm can also be implemented in time by "long division" method. Hence we can state that the Berlekamp Welch algorithm can be used to uniquely decode any Reed–Solomon code in time for a maximum of errors.
Consider a simple example where a redundant set of points are used to represent the line , and one of the points is incorrect. The points that the algorithm gets as an input are , where is the defective point. The algorithm must solve the following system of equations:
Given a solution pair to this system of equations, it is evident that at any of the points one of the following must be true:
Since is defined as only having a degree of one, the former can only be true in one point. Therefore, at the three other points.
Letting and we can rewrite the system:
This system can be solved through Gaussian elimination, and gives the values:
fits three of the four points given, so it is the most likely to be the original polynomial.