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A Single-Bit Redundancy Framework for Multi-Dimensional Parametric Constraints

Daniella Bar-Lev, Michael Shlizerman

TL;DR

This work tackles multidimensional parametric constrained coding by extending a proven universal iterative encoder-decoder framework to $d$-dimensional binary arrays of size $n^d$ with a single redundancy bit. The authors formalize parametric constraints, define encoding/decoding procedures, and demonstrate a general approach that uses an indicator and an injective mapping to iteratively fix invalid arrays. They instantiate the framework for three notable sub-array constraints—Zero-Rectangular-Cuboid-Free (ZRCF), Repeat-Free (RF), and Hamming-Distance Repeat-Free (HDRF)—achieving matches or improvements over prior bounds and, in the HDRF case, solving previously open constraints. The results highlight the framework’s versatility and potential for enabling efficient, provable constrained coding for advanced storage and communication systems, with practical implications for multidimensional data layouts and array-based media.

Abstract

Constrained coding plays a key role in optimizing performance and mitigating errors in applications such as storage and communication, where specific constraints on codewords are required. While non-parametric constraints have been well-studied, parametric constraints, which depend on sequence length, have traditionally been tackled with ad hoc solutions. Recent advances have introduced unified methods for parametric constrained coding. This paper extends these approaches to multidimensional settings, generalizing an iterative framework to efficiently encode arrays subject to parametric constraints. We demonstrate the application of the method to existing and new constraints, highlighting its versatility and potential for advanced storage systems.

A Single-Bit Redundancy Framework for Multi-Dimensional Parametric Constraints

TL;DR

This work tackles multidimensional parametric constrained coding by extending a proven universal iterative encoder-decoder framework to -dimensional binary arrays of size with a single redundancy bit. The authors formalize parametric constraints, define encoding/decoding procedures, and demonstrate a general approach that uses an indicator and an injective mapping to iteratively fix invalid arrays. They instantiate the framework for three notable sub-array constraints—Zero-Rectangular-Cuboid-Free (ZRCF), Repeat-Free (RF), and Hamming-Distance Repeat-Free (HDRF)—achieving matches or improvements over prior bounds and, in the HDRF case, solving previously open constraints. The results highlight the framework’s versatility and potential for enabling efficient, provable constrained coding for advanced storage and communication systems, with practical implications for multidimensional data layouts and array-based media.

Abstract

Constrained coding plays a key role in optimizing performance and mitigating errors in applications such as storage and communication, where specific constraints on codewords are required. While non-parametric constraints have been well-studied, parametric constraints, which depend on sequence length, have traditionally been tackled with ad hoc solutions. Recent advances have introduced unified methods for parametric constrained coding. This paper extends these approaches to multidimensional settings, generalizing an iterative framework to efficiently encode arrays subject to parametric constraints. We demonstrate the application of the method to existing and new constraints, highlighting its versatility and potential for advanced storage systems.
Paper Structure (8 sections, 2 theorems, 18 equations, 3 figures)

This paper contains 8 sections, 2 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Universal Framework for Multi-Dimensional Constrained Codes
  4. Constrained Sub-Arrays
  5. Zero-Rectangular-Cuboid-Free (ZRCF)
  6. Repeat-Free
  7. Hamming Distance Repeat-Free
  8. Discussion

Key Result

Theorem 1

Given a parametric constraint $\mathcal{C}(n,d)\subseteq \Sigma^{n^d}$, there exists an encoder-decoder pair for $\mathcal{C}(n,d)$ with a single redundancy bit if the following exist:

Figures (3)

  • Figure 1: Illustration of an array $A\in\Sigma^{5^3}$ (i.e., $n=5$ and $d=3$), with a sub-array $A_{\mathbf{I},\mathbf{d}}$, starting at position $\mathbf{I} = (0,2,2)$ and size $\mathbf{d}=(1,3,2)$. The starting position $\mathbf{I}$ is highlighted with dark blue and the rest of the sub-array is highlighted with light blue.
  • Figure 2: Illustration of the vectorization process for an array $A \in \Sigma^{3^3}$ into a one-dimensional array, the deletion of a sub-array, and reconstruction into an almost-complete array. (a) The original array $A$ with the sub-array to be deleted highlighted in blue. (b) The vectorized representation of $A$, showing the erasure of the blue sub-array. (c) The almost-complete array reconstructed from the one-dimensional vector following the deletion process.
  • Figure 3: Illustration of the process described in Lemma \ref{['lem: rf']}

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Lemma 1