Optimal prefix codes for pairs of geometrically-distributed random variables

Lossless compression is studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameter q, 0 < q < 1. Optimal prefix codes are described for q = 1/2 k (k > 1) and q = 1/knthroot2 (k > 0). These codes retain some of the low-complexity and low-latency advantage of symbol by symbol coding of geometric distributions, which is widely used in practice, while improving on the inherent redundancy of the approach. From a combinatorial standpoint, the codes described differ from previously characterized cases related to the geometric distribution in that their corresponding trees are of unbounded width, and in that an infinite set of distinct optimal codes is required to cover any interval (0,epsi), epsi > 0, of values of q