  A locally repairable code with availability has the property that every code
symbol can be recovered from multiple, disjoint subsets of other symbols of
small size. In particular, a code symbol is said to have $(r,t)$-availability
if it can be recovered from $t$ disjoint subsets, each of size at most $r$. A
code with availability is said to be 'rate-optimal', if its rate is maximum
among the class of codes with given locality, availability, and alphabet size.
  This paper focuses on rate-optimal binary, linear codes with small
availability, and makes four contributions. First, it establishes tight upper
bounds on the rate of binary linear codes with $(r,2)$ and $(2,3)$
availability. Second, it establishes a uniqueness result for binary
rate-optimal codes, showing that for certain classes of binary linear codes
with $(r,2)$ and $(2,3)$-availability, any rate optimal code must be a direct
sum of shorter rate optimal codes. Third, it presents novel upper bounds on the
rates of binary linear codes with $(2,t)$ and $(r,3)$-availability. In
particular, the main contribution here is a new method for bounding the number
of cosets of the dual of a code with availability, using its covering
properties. Finally, it presents a class of locally repairable linear codes
associated with convex polyhedra, focusing on the codes associated with the
Platonic solids. It demonstrates that these codes are locally repairable with
$t = 2$, and that the codes associated with (geometric) dual polyhedra are
(coding theoretic) duals of each other.
