  The two-dimensional magnetic recording (TDMR) technology promises storage
densities of $10$ terabits per square inch. However, when tracks are squeezed
together, a bit stored in the two-dimensional (TD) grid suffers inter-symbol
interference (ISI) from adjacent bits in the same track, and inter-track
interference (ITI) from nearby bits in the adjacent tracks. A bit is highly
likely to be read incorrectly if it is isolated in the middle of a $3 \times3$
square; surrounded by its complements, horizontally and vertically. We improve
the reliability of TDMR systems by designing two-dimensional constrained codes
that prevent these square isolation patterns. We exploit the way TD read heads
operate to design our codes, and we focus on TD read heads that collect signals
from three adjacent tracks. We represent the two-dimensional square isolation
constraint as a one-dimensional constraint on an alphabet of eight non-binary
symbols. We use this new representation to construct a non-binary
lexicographically-ordered constrained code where one third of the information
bits are unconstrained. Our TD constraint codes are capacity-achieving, and the
data protection is achieved with redundancy less than $3\%$ and at modest
complexity.
