Berlekamp–Welch algorithm

The Berlekamp–Welch algorithm, also known as the Welch–Berlekamp algorithm, is named for Elwyn R. Berlekamp and Lloyd R. Welch. This is a decoder algorithm that efficiently corrects errors in Reed–Solomon codes for an RS(n, k), code based on the Reed Solomon original view where a message ${\displaystyle m_{1},\cdots ,m_{k}}$ is used as coefficients of a polynomial ${\displaystyle F(a_{i})}$ or used with Lagrange interpolation to generate the polynomial ${\displaystyle F(a_{i})}$ of degree < k for inputs ${\displaystyle a_{1},\cdots ,a_{k}}$ and then ${\displaystyle F(a_{i})}$ is applied to ${\displaystyle a_{k+1},\cdots ,a_{n}}$ to create an encoded codeword ${\displaystyle c_{1},\cdots ,c_{n}}$.

The goal of the decoder is to recover the original encoding polynomial ${\displaystyle F(a_{i})}$, using the known inputs ${\displaystyle a_{1},\cdots ,a_{n}}$ and received codeword ${\displaystyle b_{1},\cdots ,b_{n}}$ with possible errors. It also computes an error polynomial ${\displaystyle E(a_{i})}$ where ${\displaystyle E(a_{i})=0}$ corresponding to errors in the received codeword.