Optimal Prefix Codes for Pairs of Geometrically Distributed Random Variables

Optimal prefix codes are studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameter q, 0 <; q <; 1. By encoding pairs of symbols, it may be possible to reduce the redundancy penalty of symbol-by-symbol encoding, while preserving the simplicity of the encoding and decoding procedures typical of Golomb codes and their variants. It is shown that optimal codes for these so-called two-dimensional (2-D) geometric distributions are parameter singular, in the sense that a prefix code that is optimal for one value of the parameter q cannot be optimal for any other value of q. This is in sharp contrast to the one-dimensional (1-D) case, where codes are optimal for positive-length intervals of the parameter q. Thus, in the 2-D case, it is infeasible to give a compact characterization of optimal codes for all values of the parameter q, as was done in the 1-D case. Instead, optimal codes are characterized for a discrete sequence of values of q that provides good coverage of the unit interval. Specifically, optimal prefix codes are described for q = 2^-1/k (k ≥ 1), covering the range q ≥ [1/2], and q = 2^-k (k > 1), covering the range q <; [1/2]. The described codes produce the expected reduction in redundancy with respect to the 1-D case, while maintaining low-complexity coding operations.