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Channel Coding

Module by: Behnaam Aazhang

Summary: A description of channel coding, in particular linear block codes.

Channel coding is a viable method to reduce information rate through the channel and increase reliability. This goal is achieved by adding redundancy to the information symbol vector resulting in a longer coded vector of symbols that are distinguishable at the output of the channel. Another brief explanation of channel coding is offered in Channel Coding and the Repetition Code. We consider only two classes of codes, block codes and convolutional codes.

Block codes

The information sequence is divided into blocks of length k k. Each block is mapped into channel inputs of length n n. The mapping is independent from previous blocks, that is, there is no memory from one block to another.
Example 1 
k=2 k 2 and n=5 n 5
00→00000 00 00000 (1)
01→10100 01 10100 (2)
10→01111 10 01111 (3)
11→11011 11 11011 (4)
information sequence ⇒ codeword (channel input)
A binary block code is completely defined by 2k 2 k binary sequences of length n n called codewords.
C=c1c2…c 2 k C c 1 c 2 … c 2 k (5)
ci∈01n c i 0 1 n (6)
There are three key questions,
  1. How can one find "good" codewords?
  2. How can one systematically map information sequences into codewords?
  3. How can one systematically find the corresponding information sequences from a codeword, i.e., how can we decode?
These can be done if we concentrate on linear codes and utilize finite field algebra.
A block code is linear if ci∈C c i C and cj∈C c j C implies ci⊕cj∈C c i c j C where ⊕ ⊕ is an elementwise modulo 2 addition.
Hamming distance is a useful measure of codeword properties
d H cicj=# of places that they are different d H c i c j # of places that they are different (7)
Denote the codeword for information sequence e1=1000⋮00 e 1 1 0 0 0 ⋮ 0 0 by g 1 g 1 and e2=0100⋮00 e 2 0 1 0 0 ⋮ 0 0 by g 2 g 2 ,…, and ek=0000⋮01 e k 0 0 0 0 ⋮ 0 1 by g k g k . Then any information sequence can be expressed as
u= u 1 ⋮ u k =∑i=1k u i ei u u 1 ⋮ u k i 1 k u i e i (8)
and the corresponding codeword could be
c=∑i=1k u i gi c i 1 k u i g i (9)
Therefore
c=uG c u G (10)
with c=01n c 0 1 n and u∈01k u 0 1 k where G=g1g2⋮gk G g 1 g 2 ⋮ g k , a k kxn n matrix and all operations are modulo 2.
Example 2 
In Example 1 with
00→00000 00 00000 (11)
01→10100 01 10100 (12)
10→01111 10 01111 (13)
11→11011 11 11011 (14)
g1=01111T g 1 0 1 1 1 1 and g2=10100T g 2 1 0 1 0 0 and G=0111110100 G 0 1 1 1 1 1 0 1 0 0
Additional information about coding efficiency and error are provided in Block Channel Coding.
Examples of good linear codes include Hamming codes, BCH codes, Reed-Solomon codes, and many more. The rate of these codes is defined as kn k n and these codes have different error correction and error detection properties.

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