Higher-Order Staircase Codes

TL;DR

This work generalizes staircase and zipper codes by coupling difference triangle sets (DTSs) with finite-geometric nets to form higher-order staircase codes, where each coded symbol can be protected by $M+1$ component codewords and intersections between component codewords are tightly controlled. The construction is grounded in a triad of objects—an $(L,M)$-DTS, an $(M{+}1,S)$-net, and a rate-$1{-}r/S$ component code—yielding a spatially-coupled, 4-cycle-free Tanner graph with scalable memory, encoding, and decoding characteristics. A key contribution is memory-optimal realizations via minimum-scope and minimum-sum DTSs, along with systematic, LUT-free decoding for extended Hamming components and explicit EDM-style network constructions that recover known code families (classical staircase, tiled diagonal zipper, OFEC, and MC-TDZC) as special cases. The paper also develops theoretical and computational tools to search for optimal DTSs, analyzes memory-reduction limits for higher-order variants, and demonstrates promising performance–complexity–latency tradeoffs for high-throughput, fiber-optic applications. Overall, the framework unifies several high-throughput code families and offers practical pathways to memory-efficient, low-error-floor, high-rate codes through DTS/Nets-based design and syndrome-domain decoding.

Abstract

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.

Figures (7)

• Figure 1: Visualization of a (classical) staircase code smith which corresponds to $M = 1$, $d_1=1$, and $\Pi_1(B)=B^\mathsf{T}$ (the matrix transpose); rows belong to $\mathcal{C}$.
• Figure 2: Visualization of a (weak) generalized staircase code which corresponds to $M = 2$, $(d_1,d_2)=(1,3)$, $\Pi_1(B)=B^\mathsf{T}$ (the matrix transpose), and $\Pi_2(B)=B^\pi$ (some permutation); rows belong to $\mathcal{C}$.
• Figure 3: Simulation results for extended-Hamming-based generalized staircase codes with parameters given in Table \ref{['sim-param-table']}.
• Figure 4: Visualization of a tiled diagonal zipper code corresponding to $L = 2$, $M = 1$, $(d_1^{(0)},d_1^{(1)}) = (1,2)$, and $\Pi_1(B)=B^\mathsf{T}$; rows belong to $\mathcal{C}$.
• Figure 5: Visualization of the code of Example \ref{['hosc-example']} where $L = 2$ and $M = 2$; the respective sets of distances of the two Golomb rulers are disjoint and rows belong to $\mathcal{C}$.

Theorems & Definitions (38)

• Definition 1: Ruler
• Definition 2: Weak generalized staircase code
• Definition 3: Scattering/$4$-cycle-freeness/self-orthogonality
• Proposition 1
• proof
• Remark 1
• Definition 4: $(M+1,S)$-net Colbourn
• Remark 2
• Proposition 2: Complete characterization of linear-algebraic permutations defining $(M+1,S)$-nets
• proof