  We generalize staircase codes and tiled diagonal zipper codes, preserving
their key properties while allowing each coded symbol to be protected by
arbitrarily many component codewords rather than only two. This generalization
which we term "higher-order staircase codes" arises from the marriage of two
distinct combinatorial objects: difference triangle sets and finite-geometric
nets, which have typically been applied separately to code design. We
demonstrate one possible realization of these codes, obtaining powerful,
high-rate, low-error-floor, and low-complexity coding schemes based on simple
iterative syndrome-domain decoding of coupled Hamming component codes. We
anticipate that the proposed codes could improve
performance--complexity--latency tradeoffs in high-throughput communications
applications, most notably fiber-optic, in which classical staircase codes and
zipper codes have been applied. We consider the construction of difference
triangle sets having minimum scope and sum-of-lengths, which lead to
memory-optimal realizations of higher-order staircase codes. These results also
enable memory reductions for early families of convolutional codes constructed
from difference triangle sets.
