  A toric code, introduced by Hansen to extend the Reed-Solomon code as a
$k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety
or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P
\cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are
two relevant parameters that determine the quality of a code: the information
rate, which measures how much information is contained in a single bit of each
codeword; and the relative minimum distance, which measures how many errors can
be corrected relative to how many bits each codeword has. Soprunov and
Soprunova defined a good infinite family of codes to be a sequence of codes of
unbounded polytope dimension such that neither the corresponding information
rates nor relative minimum distances go to 0 in the limit. We examine different
ways of constructing families of codes by considering polytope operations such
as the join and direct sum. In doing so, we give conditions under which no good
family can exist and strong evidence that there is no such good family of
codes.
