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 is 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 geometric distributions are \emph{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 case, where codes are optimal for positive-length intervals of
the parameter q. Thus, in the two-dimensional case, it is infeasible to give
a compact characterization of optimal codes for all values of the parameter
q, as was done in the one-dimensional case. Instead, optimal codes are
characterized for a discrete sequence of values of q that provide 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 one-dimensional case, while
maintaining low complexity coding operations.Comment: To appear in IEEE Transactions on Information Theor