Optimal prefix codes for some families of two-dimensional geometric distributions

Lossless compression is studied for pairs of independent integer-valued symbols emitted by a source with a geometric probability distribution of parameter q ∈ (0,1). Optimal prefix codes are described for q = 1/2^k (k > 1) and q = 1/k√2 (k > 0). 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, ε), ε > 0, of values of q.