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The channel coding problem
One of the most important problems that Information theory has is the channel coding problem.
The goal of the channel coding problem is to send a message reliably using a noisy channel.
The following three channels for the communication problems are important for practice:
Binary symmetric channel - It transferring bits that will be "flipped" with some probability of p and otherwise is received correctly.
Binary erasure channel - The receiver either receives the bit correctly or receives a message that the bit was not received.
Additive Gaussian noise channel - Channel transmission is defined by density and Gaussian distribution of amplitude.
Repetition code
The repetition code just repeats the transmission and keep sending the same thing again and again until it's correctly received. Encoding here is just repeating the message.
Channel code
Channel code is a maximum rate, the maximum number of bits that one can send per channel use as the probability of error vanishes ε → 0.
The rate of the code is the number of messages that the code sends per channel use.
Channel capacity
The capacity of a channel is the supremum of all achievable rates
Shannon’s channel coding theorem gives a characterization of channel capacity.
The capacity of Binary symmetric channel is equal to 1 – h (δ)
The capacity of the Binary erasure channel is equal to 1 – δ.