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Redundancy-Optimal Constructions of $(1,1)$-Criss-Cross Deletion Correcting Codes with Efficient Encoding/Decoding Algorithms

Wenhao Liu, Zhengyi Jiang, Zhongyi Huang, Hanxu Hou

TL;DR

The paper tackles reliable correction of $(1,1)$-criss-cross deletions in $n\times n$ $q$-ary arrays by introducing a novel construction that combines $q$-ary Differential VT blocks with 1-RLL Differential VT building blocks and additional modular constraints. It provides complete encoding, decoding, and data-recovery algorithms with $O(n^2)$ complexity, and proves that both code redundancy and encoder redundancy achieve $2n + 2\log_q n + O(1)$ when $q = \Omega(n)$, matching the lower bound within an $O(1)$ gap. This work delivers the first explicit construction achieving near-optimal redundancies while maintaining practical encoding/decoding procedures for all $(1,1)$-criss-cross deletion patterns. The results have potential impact for QR codes, DNA storage, and racetrack memories where robust two-dimensional deletion corrections are crucial.

Abstract

Two-dimensional error-correcting codes, where codewords are represented as $n \times n$ arrays over a $q$-ary alphabet, find important applications in areas such as QR codes, DNA-based storage, and racetrack memories. Among the possible error patterns, $(t_r,t_c)$-criss-cross deletions-where $t_r$ rows and $t_c$ columns are simultaneously deleted-are of particular significance. In this paper, we focus on $q$-ary $(1,1)$-criss-cross deletion correcting codes. We present a novel code construction and develop complete encoding, decoding, and data recovery algorithms for parameters $n \ge 11$ and $q \ge 3$. The complexity of the proposed encoding, decoding, and data recovery algorithms is $\mathcal{O}(n^2)$. Furthermore, we show that for $n \ge 11$ and $q = Ω(n)$ (i.e., there exists a constant $c>0$ such that $q \ge cn$), both the code redundancy and the encoder redundancy of the constructed codes are $2n + 2\log_q n + \mathcal{O}(1)$, which attain the lower bound ($2n + 2\log_q n - 3$) within an $\mathcal{O}(1)$ gap. To the best of our knowledge, this is the first construction that can achieve the optimal redundancy with only an $\mathcal{O}(1)$ gap, while simultaneously featuring explicit encoding and decoding algorithms.

Redundancy-Optimal Constructions of $(1,1)$-Criss-Cross Deletion Correcting Codes with Efficient Encoding/Decoding Algorithms

TL;DR

The paper tackles reliable correction of -criss-cross deletions in -ary arrays by introducing a novel construction that combines -ary Differential VT blocks with 1-RLL Differential VT building blocks and additional modular constraints. It provides complete encoding, decoding, and data-recovery algorithms with complexity, and proves that both code redundancy and encoder redundancy achieve when , matching the lower bound within an gap. This work delivers the first explicit construction achieving near-optimal redundancies while maintaining practical encoding/decoding procedures for all -criss-cross deletion patterns. The results have potential impact for QR codes, DNA storage, and racetrack memories where robust two-dimensional deletion corrections are crucial.

Abstract

Two-dimensional error-correcting codes, where codewords are represented as arrays over a -ary alphabet, find important applications in areas such as QR codes, DNA-based storage, and racetrack memories. Among the possible error patterns, -criss-cross deletions-where rows and columns are simultaneously deleted-are of particular significance. In this paper, we focus on -ary -criss-cross deletion correcting codes. We present a novel code construction and develop complete encoding, decoding, and data recovery algorithms for parameters and . The complexity of the proposed encoding, decoding, and data recovery algorithms is . Furthermore, we show that for and (i.e., there exists a constant such that ), both the code redundancy and the encoder redundancy of the constructed codes are , which attain the lower bound () within an gap. To the best of our knowledge, this is the first construction that can achieve the optimal redundancy with only an gap, while simultaneously featuring explicit encoding and decoding algorithms.
Paper Structure (18 sections, 10 theorems, 58 equations, 1 figure, 5 algorithms)

This paper contains 18 sections, 10 theorems, 58 equations, 1 figure, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $q$-ary Differential VT Codes
  4. $(1,1)$-Criss-Cross Deletion Correcting Codes
  5. Construction and Decoding Algorithm
  6. Construction
  7. Decoding Algorithm
  8. One-Dimensional 1-RLL Differential VT Code with Suffix Constraint
  9. The $q$-ary 1-RLL Differential VT Code with Suffix Constraint
  10. Encoding Algorithm of Code $\mathrm{1\text{-}RLL\_Diff\_VT}_a(n,m; \bm{b})$
  11. Decoding and Data Recovery Algorithms of Code $\mathrm{1\text{-}RLL\_Diff\_VT}_a(n,m; \bm{b})$
  12. Encoding and Data Recovery Algorithms
  13. Encoding Algorithm
  14. Data Recovery Algorithm
  15. Redundancy Analysis
  16. ...and 3 more sections

Key Result

Lemma 1

2024Differential_VT_codes Given $n\ge1$, $q\ge2$, and $a\in \mathbb{Z}_{qn}$, for any $\bm{x}\in \mathrm{Diff\_VT}_{a}(n;q)$, it holds that $\sum_{i=1}^n x_i \equiv a ~({\rm mod~}q).$

Figures (1)

  • Figure 1: Illustration of the structure of the proposed $q$-ary $(1,1)$-criss-cross deletion correcting code. The first row and the reversed last column are encoded using differential VT codes, and specific entries and modular sum constraints are imposed to ensure unique recovery from any $(1,1)$-criss-cross deletion.

Theorems & Definitions (17)

  • Lemma 1
  • Theorem 2
  • Proposition 3
  • Lemma 4
  • Theorem 5
  • Example 1
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 7 more