[Paper Review] Non-binary Codes for Correcting a Burst of at Most t Deletions

Shuche Wang, Yuanyuan Tang+3 more|arXiv (Cornell University)|Oct 21, 2022

DNA and Biological Computing5 citations

TL;DR

This paper presents new non-binary codes and permutation codes that correct bursts of up to $ t $ deletions in sequences over $ q $-ary alphabets, leveraging Varshamov-Tenengol'ts constraints and syndrome-based localization. The key contribution is the construction of codes with redundancy $ \log n + O(\log q \log \log n) $ for non-binary alphabets and $ \log n + O(\log \log n) $ for permutations, achieving the largest known code sizes for their regimes.

ABSTRACT

The problem of correcting deletions has received significant attention, partly because of the prevalence of these errors in DNA data storage. In this paper, we study the problem of correcting a consecutive burst of at most $t$ deletions in non-binary sequences. We first propose a non-binary code correcting a burst of at most 2 deletions for $q$-ary alphabets. Afterwards, we extend this result to the case where the length of the burst can be at most $t$ where $t$ is a constant. Finally, we consider the setup where the sequences that are transmitted are permutations. The proposed codes are the largest known for their respective parameter regimes.

Motivation & Objective

• To design non-binary codes capable of correcting a burst of up to $ t $ deletions in $ q $-ary sequences, where $ t $ is a constant.
• To extend existing binary burst deletion-correcting codes to non-binary alphabets using Varshamov-Tenengol'ts constraints and syndrome-based techniques.
• To construct permutation codes that correct bursts of up to $ t $ deletions with low redundancy, motivated by DNA storage and file synchronization.
• To achieve near-optimal redundancy in code size, improving upon prior constructions in terms of asymptotic efficiency.

Proposed method

• An alternative proof of Levenshtein’s binary code is developed using the Varshamov-Tenengol’ts constraint, enabling extension to non-binary alternating sequences.
• A syndrome-based localization scheme is used to confine the burst deletion to a short interval of length $ P = t2^{2t+2}\lceil\log n\rceil $, reducing the error correction problem to a smaller subproblem.
• For non-binary codes, the method employs block-wise encoding using $ E_{tB} $ functions and modular arithmetic to protect even- and odd-positioned blocks separately.
• The construction uses a carefully chosen modulus $ a = 2^{\lceil\log q\rceil(4t\log(2P)+o(\log P))} $ to ensure unique decoding of burst edits within localized intervals.
• For permutation codes, the method combines a localizing code $ \mathcal{C}_{loc}^{P} $ with a global code $ \mathcal{C}_{2t} $ that corrects overlapping ranking sequences in the localized region.
• The final code structure ensures that deletions are confined to a known interval, allowing efficient correction via rank-based reconstruction.

Experimental results

Research questions

• RQ1 Can the Varshamov-Tenengol’ts constraint be used to construct non-binary codes correcting bursts of up to 2 deletions in alternating sequences?
• RQ2 What is the minimal redundancy required for non-binary codes to correct bursts of up to $ t $ deletions over $ q $-ary alphabets?
• RQ3 Can permutation codes be constructed with redundancy $ \log n + O(\log \log n) $ to correct bursts of up to $ t $ deletions?
• RQ4 How can syndrome-based localization techniques be adapted to reduce the complexity of burst deletion correction in non-binary and permutation sequences?

Key findings

• The proposed non-binary code for correcting bursts of up to 2 deletions achieves redundancy $ \log n + O(\log q \log \log n) $, improving upon prior constructions.
• For general $ t $, the non-binary code construction achieves redundancy $ \log n + O(\log q \log \log n) $, which is the smallest known for this parameter regime.
• The permutation code for correcting bursts of up to $ t $ deletions has redundancy $ \log n + O(\log \log n) $, making it the largest known code of its type.
• The code size of the permutation code satisfies $ |\mathcal{P}_{\leq t}(n)| \geq \frac{n!}{16n \cdot 2^{O(\log \log n)}} $, indicating near-optimality in asymptotic size.
• The construction is the first to achieve $ \log n + O(\log \log n) $ redundancy for permutation codes correcting bursts of up to $ t $ deletions, matching the best-known bounds for binary cases.
• The method is robust for constant $ t $, and the redundancy remains sub-logarithmic in $ n $, making it suitable for large-scale DNA storage applications.