Error-Correcting Codes for Noisy Duplication Channels

• Published in: IEEE transactions on information theory Vol. 67; no. 6; pp. 3452 - 3463
• Main Authors: Tang, Yuanyuan, Farnoud, Farzad
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic properties; Codes; Data storage; DNA; DNA storage; Error correcting codes; Error correction; Error correction codes; exact tandem duplication; Hamming distance; Inserts; Media; Memory; Noise measurement; noisy tandem duplication; Reproduction (copying); Transforms
• DOI: 10.1109/TIT.2021.3059095
• ISSN: 0018-9448

Because of its high data density and longevity, DNA is emerging as a promising candidate for satisfying increasing data storage needs. Compared to conventional storage media, however, data stored in DNA is subject to a wider range of errors resulting from various processes involved in the data storage pipeline. In this article, we consider correcting duplication errors for both exact and noisy tandem duplications of a given length <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>. An exact duplication inserts a copy of a substring of length <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> of the sequence immediately after that substring, e.g., <inline-formula> <tex-math notation="LaTeX">\mathsf {ACGT} \to \mathsf {ACG\underline {ACG}T} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">k=3 </tex-math></inline-formula>, while a noisy duplication inserts a copy suffering from substitution noise, e.g., <inline-formula> <tex-math notation="LaTeX">\mathsf {ACGT} \to \mathsf {ACG\underline {A \color {Red}{T}}GT} </tex-math></inline-formula>. Specifically, we design codes that can correct any number of exact duplication and one noisy duplication errors, where in the noisy duplication case the copy is at Hamming distance 1 from the original. Our constructions rely upon recovering the duplication root of the stored codeword. We characterize the ways in which duplication errors manifest in the root of affected sequences and design efficient codes for correcting these error patterns. We show that the proposed construction is asymptotically optimal, in the sense that it has the same asymptotic rate as optimal codes correcting exact duplications only.