Question

What are channel coding theorem for noiseless channel?

Answer 1

1. For every discrete memoryless channel, the channel capacity
{\displaystyle \ C=\sup _{p_{X}}I(X;Y)}[2]
has the following property. For any ε > 0 and R < C, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε.
2. If a probability of bit error pb is acceptable, rates up to R(pb) are achievable, where
R(pb)=C/1-H2(Pb)
Nd H2(Pb)= -[pb log2pb +(1-pblog)]
{\displaystyle R(p_{b})={\frac {C}{1-H_{2}(p_{b})}}.}
and {\displaystyle H_{2}(p_{b})} is the binary entropy function
{\displaystyle H_{2}(p_{b})=-\left[p_{b}\log _{2}{p_{b}}+(1-p_{b})\log _{2}({1-p_{b}})\right]}
3. For any pb, rates greater than R(pb) are not achievable.