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LOCO Codes Can Correct as Well: Error-Correction Constrained Coding for DNA Data Storage

Canberk İrimağzı, Ahmed Hareedy

TL;DR

This work extends DNA LOCO codes to EC D-LOCO codes that correct substitution errors via syndrome-like residue decoding, using a redundancy metric $R(m)$ to select indices divisible by $R(m)$ under run-length constraints $\ell \in \{1,2,3\}$ and GC-content balancing. It introduces a 3-symbol bridging scheme, a list-decoding extension for double substitutions, and a detailed analysis of index differences to bound the residue-table size, showing that the redundancy scales as $2\log_2(m)+O(1)$ and the rate is capacity-approaching for the constrained system. The paper provides a formal construction of $\mathcal{D}^{\mathsf{Res}}_{m,\ell}$, a residue decoding algorithm with $O(m^2)$ to $O(m^3)$ complexity, and comparative assessments against existing schemes, along with modeling of $R(m)$ and practical storage implications. The proposed approach enables scalable, reliable DNA data storage with constrained runs and GC balance, while supporting single-substitution correction and high-probability double-substitution correction, paving the way for reconstruction-assisted handling of deletions and insertions in future work.

Abstract

As a medium for cold data storage, DNA stands out as it promises significant gains in storage capacity and lifetime. However, it comes with its own data processing challenges to overcome. Constrained codes over the DNA alphabet $\{A,T,G,C\}$ have been used to design DNA sequences that are free of long homopolymers to increase stability, yet effective error detection and error correction are required to achieve reliability in data retrieval. Recently, we introduced lexicographically-ordered constrained (LOCO) codes, namely DNA LOCO (D-LOCO) codes, with error detection. In this paper, we equip our D-LOCO codes with error correction for substitution errors via syndrome-like decoding, designated as residue decoding. We only use D-LOCO codewords of indices divisible by a suitable redundancy metric $R(m) > 0$, where $m$ is the code length, for error correction. We provide the community with a construction of constrained codes forbidding runs of length higher than fixed $\ell \in \{1,2,3\}$ and $GC$-content in $\big [0.5-\frac{1}{2K},0.5+\frac{1}{2K}\big ]$ that correct $K$ segmented substitution errors, one per codeword. We call the proposed codes error-correction (EC) D-LOCO codes. We also give a list-decoding procedure with near-quadratic time-complexity in $m$ to correct double-substitution errors within EC D-LOCO codewords, which has $> 98.20\%$ average success rate. The redundancy metric is projected to require $2\log_2(m)+O(1)$-bit allocation for a length-$m$ codeword. Hence, our EC D-LOCO codes are projected to be capacity-approaching with respect to the error-free constrained system.

LOCO Codes Can Correct as Well: Error-Correction Constrained Coding for DNA Data Storage

TL;DR

This work extends DNA LOCO codes to EC D-LOCO codes that correct substitution errors via syndrome-like residue decoding, using a redundancy metric to select indices divisible by under run-length constraints and GC-content balancing. It introduces a 3-symbol bridging scheme, a list-decoding extension for double substitutions, and a detailed analysis of index differences to bound the residue-table size, showing that the redundancy scales as and the rate is capacity-approaching for the constrained system. The paper provides a formal construction of , a residue decoding algorithm with to complexity, and comparative assessments against existing schemes, along with modeling of and practical storage implications. The proposed approach enables scalable, reliable DNA data storage with constrained runs and GC balance, while supporting single-substitution correction and high-probability double-substitution correction, paving the way for reconstruction-assisted handling of deletions and insertions in future work.

Abstract

As a medium for cold data storage, DNA stands out as it promises significant gains in storage capacity and lifetime. However, it comes with its own data processing challenges to overcome. Constrained codes over the DNA alphabet have been used to design DNA sequences that are free of long homopolymers to increase stability, yet effective error detection and error correction are required to achieve reliability in data retrieval. Recently, we introduced lexicographically-ordered constrained (LOCO) codes, namely DNA LOCO (D-LOCO) codes, with error detection. In this paper, we equip our D-LOCO codes with error correction for substitution errors via syndrome-like decoding, designated as residue decoding. We only use D-LOCO codewords of indices divisible by a suitable redundancy metric , where is the code length, for error correction. We provide the community with a construction of constrained codes forbidding runs of length higher than fixed and -content in that correct segmented substitution errors, one per codeword. We call the proposed codes error-correction (EC) D-LOCO codes. We also give a list-decoding procedure with near-quadratic time-complexity in to correct double-substitution errors within EC D-LOCO codewords, which has average success rate. The redundancy metric is projected to require -bit allocation for a length- codeword. Hence, our EC D-LOCO codes are projected to be capacity-approaching with respect to the error-free constrained system.

Paper Structure

This paper contains 21 sections, 2 theorems, 36 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Preliminaries
  4. Index Errors Due to Single Substitution for $\ell =1$
  5. Index Differences for $\ell =1$
  6. Changes in Symbol Contributions for $\ell =1$
  7. Index Errors Due to Single Substitution for $\ell=2$
  8. Index Differences for $\ell =2$
  9. Changes in Symbol Contributions for $\ell =2$
  10. Interdependence of Symbol Contributions for $\ell=2$
  11. Residue Decoding
  12. Bridging Scheme and Balancing
  13. Steps of the Residue Decoding Algorithm
  14. Storage Overhead and Capacity Approachability
  15. Achieving Optimal Storage for Adopted Redundancy
  16. ...and 6 more sections

Key Result

Theorem 1

The encoding-decoding rule $g: \mathcal{D}_{m,\ell} \rightarrow \{0,1,\newline\dots,N(m)-1\}$, for general $\ell \geq 1$, is as follows: For $1 \leq k \leq \ell$, $\delta_{i,k} \in \{0,1\}$ and $\delta_ {i,k} = 1$ if and only if $c_i > \Delta, c_{i+j} = \Delta$ for all $i+1 \leq j \leq i+k-1$, and $c_{i+k} \neq \Delta$. Note that for $k = 1$, the condition $c_{i+j} = \Delta$ for all $i+1 \leq j \

Figures (2)

  • Figure 2: Projected rates of EC D-LOCO codes $\mathcal{D}^{\mathsf{Res}}_{m,2}$ and $\mathcal{D}^{\mathsf{Res}}_{m,3}$ based on the quadratic modeling of $R(m,2)$ and $R(m,3)$. The rates on the red (blue) curve are actual values for $m \in [15,59]$ ($m \in [15,27]$).
  • Figure 3: Panorama of the data retrieval process to illustrate how our error-correction algorithm would fit into it once preceded by a reconstruction algorithm. The dashed line following the reconstruction algorithm indicates an outcome of correct length with high probability, which our decoder effectively handles.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Theorem 1
  • proof
  • Definition 4
  • Definition 5
  • Example 1
  • Remark 2
  • ...and 25 more