Lossy source code using a constrained random number generator

A stochastic encoder for lossy source coding is introduced with a rate-distortion pair close to the boundary of the rate-distortion region, where the only restriction is that the reproduction 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 rate-distortion region. Since an ensemble of sparse matrices has a hash property, we can construct a code by using sparse matrices.