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.

Just as rank-metric or Gabidulin codes may be used to construct rate–diversity tradeoff optimal space–time codes, a recently introduced generalization for the sum-rank metric—linearized Reed–Solomon codes—accomplishes the same in the case of multiple fading blocks. In this paper, we provide the first explicit construction of minimal delay rate–diversity optimal multiblock space–time codes as an application of linearized Reed–Solomon codes. We also provide sequential decoders for these codes and, more generally, space–time codes constructed from finite field codes. Simulation results show that the proposed codes can outperform full diversity codes based on cyclic division algebras at low SNRs as well as utilize significantly smaller constellations.

Just as rank-metric or Gabidulin codes may be used to construct rate-diversity tradeoff optimal space-time codes, a recently introduced generalization for the sum-rank metric, linearized Reed-Solomon codes, accomplishes the same in the case of multiple fading blocks. We provide the first explicit construction of minimal-delay rate-diversity optimal multiblock space-time codes as an application of linearized Reed-Solomon codes. We then demonstrate in simulation an example of a 2-block 2-by-2 code which, with a small performance penalty-less than 1 dB at a codeword error rate of 1e-4-matches the bit rate of a full diversity alternative while using a much smaller transmitted constellation. A stack decoder for this code is then suggested.