  Because of their excellent asymptotic and finite-length performance,
spatially-coupled (SC) codes are a class of low-density parity-check codes that
is gaining increasing attention. Multi-dimensional (MD) SC codes are
constructed by connecting copies of an SC code via relocations in order to
mitigate various sources of non-uniformity and improve performance in many data
storage and data transmission systems. As the number of degrees of freedom in
the MD-SC code design increases, appropriately exploiting them becomes more
difficult because of the complexity growth of the design process. In this
paper, we propose a probabilistic framework for the MD-SC code design, which is
based on the gradient-descent (GD) algorithm, to design better MD codes and
address this challenge. In particular, we express the expected number of short
cycles, which we seek to minimize, in the graph representation of the code in
terms of entries of a probability-distribution matrix that characterizes the
MD-SC code design. We then find a locally-optimal probability distribution,
which serves as the starting point of a finite-length algorithmic optimizer
that produces the final MD-SC code. We offer the theoretical analysis as well
as the algorithms, and we present experimental results demonstrating that our
MD codes, conveniently called GD-MD codes, have notably lower short cycle
numbers compared with the available state-of-the-art. Moreover, our algorithms
converge on solutions in few iterations, which confirms the complexity
reduction as a result of limiting the search space via the locally-optimal
GD-MD distributions.
