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Error Correcting Codes: Combinatorics, Algorithms and Applications
(Fall 2007)
Lecture 26: Decoding Concatenated Codes
October 29, 2007
Lecturer: Atri Rudra
Scribe: Michael Pfetsch & Atri Rudra
In the last lecture, we saw Justesen’s strongly explicit asymptotically good code. In today’s
lecture, we will begin to answer the natural question of whether we can decode concatenated codes
up to half their design distance in polynomial time.
1
Decoder for concatenated codes
The concatenation of codes, which was introduced in Lecture 24, is illustrated in Figure 1.
Figure 1: Code concatenation and unique decoding of the code
C
out
◦
C
in
.
We begin with a natural decoding algorithm for concatenated codes that “reverses” the encod
ing process. In particular, the code ﬁrst decodes the inner code and then decodes the outer code.
For the time being let us assume that we have a polynomial time unique decoding algorithm for
the outer code. This leaves us with the task of coming up with a polynomial time decoding algo
rithm for the inner codes. Our task of coming up such a decoder is made easier by the fact that
1
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View Full Documentthe polynomial running time needs to be polynomial in the
ﬁnal
block length. This in turn implies
that we would be ﬁne if we picked a decoding algorithm that runs in time singly exponential in the
inner block length as long as the inner block length is logarithmic in the outer code block length.
However, note that the latter is what we have assumed so far and thus, we can use the Maximum
Likelihood Decoder (or MLD) for the inner code.
More formally, let the input to the decoder be the vector
y
= (
y
1
,
···
,y
N
)
∈
[
q
n
]
N
. The
decoding algorithm is a two step process:
1.
Step 1
: Compute
y
0
= (
y
0
1
,
0
N
)
∈
[
q
k
]
N
as follows
y
0
i
=
MLD
C
in
(
y
i
) 1
≤
i
≤
N.
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 Spring '11
 RUDRA
 Algorithms

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