Error Correcting Codes: Combinatorics, Algorithms and Applications
(Fall 2007)
Lecture 2: Error Correction and Channel Noise
August 29, 2007
Lecturer: Atri Rudra
Scribe: Yang Wang & Atri Rudra
As was mentioned in the last lecture, the fundamental tradeoff we are interested in for this
course is the one between the amount of redundancy in the code vs. the number of errors that it
can correct. We defined the notion of rate of a code to capture the amount of redundancy. However,
before we embark on a systematic study of the tradeoff above, we need to formally define what it
means to correct errors. We do so next.
1
Error correction
Before we define what we mean by error correction, we formally define the notion of
encoding
.
Definition 1.1 (Encoding function).
Let
. An equivalent description of the code
is by
an injective mapping
called encoding function.
Next we move to error correction. Intuitively, we can correct a received word if we can recover
the transmitted codeword (or equivalently the corresponding message). This “reverse” process is
achieved by
decoding
.
Definition 1.2 (Decoding function).
Let
be a code. A mapping
is called
a decoding function for
.
The definition of a decoding function by itself does not give anything interesting. What we
really need from a decoding function is that it recovers the transmitted message. This notion is
captured next.
Definition 1.3 (Error Correction).
Let
and let
be an integer.
is said to be

errorcorrecting
if there exists a decoding function
such that for every error message
and error pattern
with at most
errors,
.
Figure 1 illustrates how the definitions we have examined so far interact.
We will also very briefly look at a weaker form of error recovery called
error detection
.
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 Spring '10
 Rejaei
 Information Theory, Electromagnet, Error detection and correction, error detection

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