On the Palindromic/Reverse-Complement Duplication Correcting Codes

TL;DR

The paper tackles correcting reverse-complement and palindromic duplications in DNA storage by exploiting duplication structure to reduce redundancy. It provides an explicit single-redundancy code that can handle arbitrarily many disjoint duplications for long copies, and a Gilbert–Varshamov bound showing redundancy as low as $2\log_q n + \log_q\log_q n + O(1)$ for arbitrary-length duplications. For length-one duplications with $q \ge 4$, it devises two constructions achieving redundancies $2t\log_q n + O(\log_q\log_q n)$ and $(2t-1)\log_q n + O(\log_q\log_q n)$, with trade-offs in encoding/decoding complexity. These results improve on general burst-insertion codes and yield practical, scalable coding schemes for robust in-vivo DNA data storage.

Abstract

Motivated by applications in in-vivo DNA storage, we study codes for correcting duplications. A reverse-complement duplication of length $k$ is the insertion of the reversed and complemented copy of a substring of length $k$ adjacent to its original position, while a palindromic duplication only inserts the reversed copy without complementation. We first construct an explicit code with a single redundant symbol capable of correcting an arbitrary number of reverse-complement duplications (respectively, palindromic duplications), provided that all duplications have length $k \ge 3\lceil \log_q n \rceil$ and are disjoint. Next, we derive a Gilbert-Varshamov bound for codes that can correct a reverse-complement duplication (respectively, palindromic duplication) of arbitrary length, showing that the optimal redundancy is upper bounded by $2\log_q n + \log_q\log_q n + O(1)$. Finally, for $q \ge 4$, we present two explicit constructions of codes that can correct $t$ length-one reverse-complement duplications. The first construction achieves a redundancy of $2t\log_q n + O(\log_q\log_q n)$ with encoding complexity $O(n)$ and decoding complexity $O\big(n(\log_2 n)^4\big)$. The second construction achieves an improved redundancy of $(2t-1)\log_q n + O(\log_q\log_q n)$, but with encoding and decoding complexities of $O\big(n \cdot \mathrm{poly}(\log_2 n)\big)$.

Figures (1)

• Figure 3.1: Illustrations of the definitions of $\bm{u}^{(t)}$ for $t\in [1,2s]$ and $\bm{v}^{(t)}$ for $t\in [1,2s-1]$.

Theorems & Definitions (60)

• Remark 2.1
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.4
• Definition 3.5
• Theorem 3.6
• proof