  In distributed storage systems that employ erasure coding, the issue of
minimizing the total {\it repair bandwidth} required to exactly regenerate a
storage node after a failure arises. This repair bandwidth depends on the
structure of the storage code and the repair strategies used to restore the
lost data. Minimizing it requires that undesired data during a repair align in
the smallest possible spaces, using the concept of interference alignment (IA).
Here, a points-on-a-lattice representation of the symbol extension IA of
Cadambe {\it et al.} provides cues to perfect IA instances which we combine
with fundamental properties of Hadamard matrices to construct a new storage
code with favorable repair properties. Specifically, we build an explicit
$(k+2,k)$ storage code over $\mathbb{GF}(3)$, whose single systematic node
failures can be repaired with bandwidth that matches exactly the theoretical
minimum. Moreover, the repair of single parity node failures generates at most
the same repair bandwidth as any systematic node failure. Our code can tolerate
any single node failure and any pair of failures that involves at most one
systematic failure.
