Optimal random coding

We consider a stochastic method for representing a real number x by a finite sequence of bits. The method is symmetrical with respect to interchange of the bits: it is characterized by a single function p(x) which gives the probability that any particular bit is on as a function of the real number x that is being encoded. We then consider the problem of reconstructing x from its representation. In the limiting case in which the number of bits is large, we determine the function p(x) that minimizes the expected k‐th power of the absolute error in this reconstruction. The optimal choice of p(x) is independent of k.