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---
title: "Berlekamp–Welch algorithm"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Berlekamp–Welch_algorithm"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T14:40:01.393171+00:00"
instance: "kb-cron"
---
q
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e
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e
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0
{\displaystyle {\begin{alignedat}{10}q_{0}&+&q_{1}&+&q_{2}&-&4e_{0}&-&4&=&0\\q_{0}&+&2q_{1}&+&4q_{2}&-&3e_{0}&-&6&=&0\\q_{0}&+&3q_{1}&+&9q_{2}&-&4e_{0}&-&12&=&0\\q_{0}&+&4q_{1}&+&16q_{2}&-&e_{0}&-&4&=&0\end{alignedat}}}
This system can be solved through Gaussian elimination, and gives the values:
q
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15
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8
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{\displaystyle q_{0}=-15,q_{1}=8,q_{2}=-1,e_{0}=-3}
Thus,
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,
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{\displaystyle Q(x)=-x^{2}+8x-15,E(x)=x-3}
. Dividing the two gives:
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{\displaystyle {Q(x) \over E(x)}=P(x)=5-x}
5
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x
{\displaystyle 5-x}
fits three of the four points given, so it is the most likely to be the original polynomial.
== Example ==
Consider RS(7,3) (n = 7, k = 3) defined in GF(7) with α = 3 and input values: ai = i-1 : {0,1,2,3,4,5,6}. The message to be systematically encoded is {1,6,3}. Using Lagrange interpolation, F(ai) = 3 x2 + 2 x + 1, and applying F(ai) for a4 = 3 to a7 = 6, results in the code word {1,6,3,6,1,2,2}. Assume errors occur at c2 and c5 resulting in the received code word {1,5,3,6,3,2,2}. Start off with e = 2 and solve the linear equations:
[
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]
[
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=
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]
{\displaystyle {\begin{bmatrix}b_{1}&b_{1}a_{1}&-1&-a_{1}&-a_{1}^{2}&-a_{1}^{3}&-a_{1}^{4}\\b_{2}&b_{2}a_{2}&-1&-a_{2}&-a_{2}^{2}&-a_{2}^{3}&-a_{2}^{4}\\b_{3}&b_{3}a_{3}&-1&-a_{3}&-a_{3}^{2}&-a_{3}^{3}&-a_{3}^{4}\\b_{4}&b_{4}a_{4}&-1&-a_{4}&-a_{4}^{2}&-a_{4}^{3}&-a_{4}^{4}\\b_{5}&b_{5}a_{5}&-1&-a_{5}&-a_{5}^{2}&-a_{5}^{3}&-a_{5}^{4}\\b_{6}&b_{6}a_{6}&-1&-a_{6}&-a_{6}^{2}&-a_{6}^{3}&-a_{6}^{4}\\b_{7}&b_{7}a_{7}&-1&-a_{7}&-a_{7}^{2}&-a_{7}^{3}&-a_{7}^{4}\\\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q0\\q1\\q2\\q3\\q4\\\end{bmatrix}}={\begin{bmatrix}-b_{1}a_{1}^{2}\\-b_{2}a_{2}^{2}\\-b_{3}a_{3}^{2}\\-b_{4}a_{4}^{2}\\-b_{5}a_{5}^{2}\\-b_{6}a_{6}^{2}\\-b_{7}a_{7}^{2}\\\end{bmatrix}}}
[
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]
[
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=
[
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{\displaystyle {\begin{bmatrix}1&0&6&0&0&0&0\\5&5&6&6&6&6&6\\3&6&6&5&3&6&5\\6&4&6&4&5&1&3\\3&5&6&3&5&6&3\\2&3&6&2&3&1&5\\2&5&6&1&6&1&6\\\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q0\\q1\\q2\\q3\\q4\\\end{bmatrix}}={\begin{bmatrix}0\\2\\2\\2\\1\\6\\5\\\end{bmatrix}}}
[
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]
[
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=
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{\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&1&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\\\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q0\\q1\\q2\\q3\\q4\\\end{bmatrix}}={\begin{bmatrix}4\\2\\4\\3\\3\\1\\3\\\end{bmatrix}}}
Starting from the bottom of the right matrix, and the constraint e2 = 1:
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=
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{\displaystyle Q(a_{i})=3x^{4}+1x^{3}+3x^{2}+3x+4}
E
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{\displaystyle E(a_{i})=1x^{2}+2x+4}
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/
E
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{\displaystyle F(a_{i})=Q(a_{i})/E(a_{i})=3x^{2}+2x+1}
with remainder = 0.
E(ai) = 0 at a2 = 1 and a5 = 4
Calculate F(a2 = 1) = 6 and F(a5 = 4) = 1 to produce corrected code word {1,6,3,6,1,2,2}.
== See also ==
Reed–Solomon error correction
== External links ==
MIT Lecture Notes on Essential Coding Theory – Dr. Madhu Sudan
University at Buffalo Lecture Notes on Coding Theory – Dr. Atri Rudra
Algebraic Codes on Lines, Planes and Curves, An Engineering Approach – Richard E. Blahut
Welch Berlekamp Decoding of Reed–Solomon Codes – L. R. Welch
US 4,633,470, Welch, Lloyd R. & Berlekamp, Elwyn R., "Error Correction for Algebraic Block Codes", published September 27, 1983, issued December 30, 1986 – The patent by Lloyd R. Welch and Elewyn R. Berlekamp
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