  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 $q=2^{-1/k}$ ($k\ge 1$), covering the range $q\ge 1/2$, 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.
