Channel code using a constrained Random number generator

A stochastic encoder for channel coding is introduced with a rate close to the channel capacity, where the only restriction is that the channel input alphabet is finite. Random numbers, which satisfy a condition specified by a function and its value, are used to construct the stochastic encoder. The proof of the theorem is based on the hash property of an ensemble of functions, where the results are extended to a general channel by deriving an alternative formula for the capacity. Since an ensemble of sparse matrices has a hash property, we can construct a code by using sparse matrices.