  Understanding the limits imposed on information storage capacity of physical
systems is a problem of fundamental and practical importance which bridges
physics and information science. There is a well-known upper bound on the
amount of information that can be stored reliably in a given volume of discrete
spin systems which are supported by gapped local Hamiltonians. However, all the
previously known systems were far below this theoretical bound, and it remained
open whether there exists a gapped spin system that saturates this bound. Here,
we present a construction of spin systems which saturate this theoretical limit
asymptotically by borrowing an idea from fractal properties arising in the
Sierpinski triangle. Our construction provides not only the best classical
error-correcting code which is physically realizable as the energy ground space
of gapped frustration-free Hamiltonians, but also a new research avenue for
correlated spin phases with fractal spin configurations.
