Error Correcting Codes
Basic Error-Correction
We will consider two situations in which we wish to transmit information on an unreliable channel. The
first is exemplified by the internet, where the information (say a file) is broken up into packets, and the
unreliability is manifest in the fact that some of the packets are lost during transmission. Suppose that the
message consists of npackets and suppose that kpackets are lost during transmission. We will show how
to encode the initial message consisting of npackets into a redundant encoding consisting of n+kpackets
such that the recipient can reconstruct the message from anynreceived packets. Note that in this setting the
packets are labelled and thus the recipient knows exactly which packets were dropped during transmission.
Assume that the contents of each packet can be represented as a number modulo qfor some prime q. The
error-correcting code that we shall use is based on polynomials over the Galois Field GF(q). Denote the
message by m1,...,mn. We will assume that qis sufficiently large so that q>n+k. Consider the unique
polynomial P(x)of degree n−1 such that P(i) = mifor 1 ≤i≤n. The encoded message is cj=P(j)
for 1 ≤j≤n+k. Using Lagrange Interpolation, the recipient can reconstruct the polynomial P(x)from
the value of P(x)at any npoints. This error-correcting scheme is therefore optimal—it can recover the n
characters of the transmitted message from any nreceived characters.
The Berlekamp-Welch Algorithm
The second situation that we shall consider is more challenging. In this situation, Alice wishes to send Bob
a message over a noisy channel. This time, instead of erasures or packet losses, some of the transmitted
packets are corrupted. What makes this situation more challenging for Bob is that he does not know which
of the received packets (characters) are corrupted and which were transmitted accurately. Polynomials still
provide an optimal scheme for dealing with this situation. Alice must transmit n+2kcharacters to enable
Bob to recover from kgeneral errors, i.e., the encoded message is now cj=P(j)for 1 ≤j≤n+2k.
The received message R(j)for 1 ≤j≤n+2kdiffers from the polynomial P(x)at kpoints. How can Bob
reconstruct P(x)from these n+2kvalues R(j)? Our first observation is that if Bob can find any polynomial
P0(x)of degree n−1 that agrees with R(x)at n+kpoints then P0(x) = P(x). This is because out of the n+k
points there are at most kerrors, and therefore on at least npoints P0(x) = P(x). But a polynomial of degree
n−1 is uniquely defined by its values at npoints.
But how do we find such a polynomial? We could try to guess where the kerrors lie, but this would take too
long (it would take exponential time, in fact). A very clever polynomial-time algorithm for this problem was
invented by Berlekamp and Welch. The main idea is to describe the received message R(x)(which because
of the errors is not a polynomial) as a ratio of polynomial. Let e1,...,ekbe the kpositions at which errors
occurred. Define the error locator polynomial E(x) = (x−e1)(x−e2)···(x−ek).E(x)is 0 at exactly the k
points at which errors occurred. Now observe that for all n+2kpoints 1 ≤x≤n+2k,P(x)E(x) = R(x)E(x).
CS 70, Fall 2004, Lecture 9 1
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