Fair noiseless broadcast source coding

We present a noiseless source coding problem in a broadcast environment and supply a simple solution to this problem. A transmitter wishes to transmit a binary random vector X₁ⁿ = (X₁, X₂, ..., X_n) to n receivers, where receiver k is only interested in the component X_k. A source encoding is a binary sequence f = (f₁, f₂, ...) which is chosen by the transmitter. The expected time at which the k^th receiver determines X_k is denoted l(f, k). This means that the initial segment (f₁, f₂, ..., f_{l(f, k)}) of the encoding allows unique decoding of X_k. We define the performance measure L(n) = min_f max^k l(f, k), where the minimization is over all possible encoding, and wish to approach it by practical schemes. For the case of independent but not necessarily identically distributed Bernoulli sources, we demonstrate encoding scheme f for which; lim _n→∞ [max_k l(f, k)/(n + 1)/2] = 1, where n+1/2 is the arithmetic mean of the values (l(f, K))_k=1ⁿ obtained by the naive scheme f_k = X_k. In the naive scheme, the worst case receiver learns its value only after n bits have been received, so this is a substantial improvement. In conclusion, we constructively establish that the inequality L(n) ≤ n+3/2 holds for stationary, ergodic and bitwise independent sources. We also generalize our results to the case where each receiver is interested in a block of data, as opposed to only one bit. The determination of flower bounds on L(n) is still open.