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Higher-Order Staircase Codes: A Unified Generalization of High-Throughput Coding Techniques

Mohannad Shehadeh, Frank R. Kschischang

TL;DR

This work introduces higher-order staircase codes as a unified framework that generalizes many high-throughput coding techniques by integrating difference triangle sets ($L,M$) with finite-geometric nets and a component code $$, achieving a rate $1 - r/S$ and a flexible memory/parallelism design via parameters $L$ and $C$. The construction yields a spatially-coupled, 4-cycle-free Tanner graph where each symbol is protected by $M+1$ component codes and any pair of codes shares at most one symbol, enabling efficient syndrome-domain iterative decoding. By recovering classical staircase codes when $L=M=C=1$ and interpolating among OFEC, CI, tiled diagonal zipper codes, MC-TDZC, and Robinson-Bernstein convolutional codes, the approach offers broad design versatility with controllable error floors and thresholds. Empirical results using extended-Hamming components on a BSC demonstrate competitive throughput with reduced memory and complexity, while theoretical sections investigate DTS with minimum scope and sum-of-lengths, including recursive constructions and infinite families, to underpin memory-efficient, scalable code designs.

Abstract

We introduce a unified generalization of several well-established high-throughput coding techniques including staircase codes, tiled diagonal zipper codes, continuously interleaved codes, open forward error correction (OFEC) codes, and Robinson-Bernstein convolutional codes as special cases. 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 illustrate 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 study some properties of difference triangle sets having minimum scope and sum-of-lengths, which correspond to memory-optimal higher-order staircase codes.

Higher-Order Staircase Codes: A Unified Generalization of High-Throughput Coding Techniques

TL;DR

This work introduces higher-order staircase codes as a unified framework that generalizes many high-throughput coding techniques by integrating difference triangle sets () with finite-geometric nets and a component code , achieving a rate and a flexible memory/parallelism design via parameters and . The construction yields a spatially-coupled, 4-cycle-free Tanner graph where each symbol is protected by component codes and any pair of codes shares at most one symbol, enabling efficient syndrome-domain iterative decoding. By recovering classical staircase codes when and interpolating among OFEC, CI, tiled diagonal zipper codes, MC-TDZC, and Robinson-Bernstein convolutional codes, the approach offers broad design versatility with controllable error floors and thresholds. Empirical results using extended-Hamming components on a BSC demonstrate competitive throughput with reduced memory and complexity, while theoretical sections investigate DTS with minimum scope and sum-of-lengths, including recursive constructions and infinite families, to underpin memory-efficient, scalable code designs.

Abstract

We introduce a unified generalization of several well-established high-throughput coding techniques including staircase codes, tiled diagonal zipper codes, continuously interleaved codes, open forward error correction (OFEC) codes, and Robinson-Bernstein convolutional codes as special cases. 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 illustrate 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 study some properties of difference triangle sets having minimum scope and sum-of-lengths, which correspond to memory-optimal higher-order staircase codes.

Paper Structure

This paper contains 5 sections, 9 theorems, 24 equations, 3 figures.

Table of Contents

  1. Overview
  2. Code Construction
  3. Simulations
  4. Difference Triangle Sets with Minimum Scope and Sum-of-Lengths
  5. Future Work

Key Result

Proposition 1

A collection of $M+1$ permutations of $\mathcal{R} \times \mathcal{R}$, where $\mathcal{R}$ is a finite commutative ring of cardinality $S/L$, defined by left-multiplication of a set of invertible $2 \times 2$ matrices, defines an $(M+1,S/L)$-net if and only if, for any pair of distinct matrices $A, we have that $c\tilde{d} - d \tilde{c}$ is invertible in $\mathcal{R}$.

Figures (3)

  • Figure 1: Visualization of the code of Example \ref{['hosc-example']} where $L = 2$ and $M = 2$; rows belong to $\mathcal{C}$ in this picture.
  • Figure 2: Simulation results for $1-r/S \approx 0.937$ examples with parameters $(L,M,S/L,C,W,I,1-r/S,WC(S/L)^2L)$.
  • Figure 3: Simulation results for $1-r/S \approx 0.87$ examples with parameters $(L,M,S/L,C,W,I,1-r/S,WC(S/L)^2L)$.

Theorems & Definitions (14)

  • Proposition 1: Complete characterization of linear transformations defining $(M+1,S/L)$-nets
  • Example 1
  • Example 2
  • Definition 1: Higher-order staircase code
  • Proposition 2
  • Corollary 2.1
  • Example 3
  • Definition 2: Sharply $2$-transitive permutation group
  • Theorem 3: Special case of the combining construction of Wild Wild and Kotzig and Turgeon Kotzig-Turgeon
  • Proposition 4: Sum-of-lengths under combining
  • ...and 4 more