Correcting a Single Indel/Edit for DNA-Based Data Storage: Linear-Time Encoders and Order-Optimality

• Published in: IEEE transactions on information theory Vol. 67; no. 6; pp. 3438 - 3451
• Main Authors: Cai, Kui, Chee, Yeow Meng, Gabrys, Ryan, Kiah, Han Mao, Nguyen, Tuan Thanh
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Alphabets; Coders; Codes; Data storage; Decoding; Deletion; DNA; DNA-based data storage; Encoding; Error correction codes; GC-balanced codes; In vivo; Insertion; Messages; Optimization; Redundancy; Sequential analysis; Signal processing algorithms
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2021.3049627

An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. In this article, we investigate codes that correct either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with <inline-formula> <tex-math notation="LaTeX">\lceil {\log \text {n}}\rceil+\text {O}(\log \log \text {n}) </tex-math></inline-formula> redundancy bits, while the other corrects a single indel with <inline-formula> <tex-math notation="LaTeX">\lceil {\log \text {n}}\rceil+2 </tex-math></inline-formula> redundant bits. These two encoders are order-optimal . The former encoder is the first known order-optimal encoder that corrects a single edit, while the latter encoder (that corrects a single indel) reduces the redundancy of the best known encoder of Tenengolts (1984) by at least four bits. Over the DNA alphabet, we impose an additional constraint: the <inline-formula> <tex-math notation="LaTeX">\mathtt {GC} </tex-math></inline-formula> -balanced constraint and require that exactly half of the symbols of any DNA codeword to be either <inline-formula> <tex-math notation="LaTeX">\mathtt {C} </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">\mathtt {G} </tex-math></inline-formula>. In particular, via a modification of Knuth's balancing technique, we provide a linear-time map that translates binary messages into <inline-formula> <tex-math notation="LaTeX">\mathtt {GC} </tex-math></inline-formula>-balanced codewords and the resulting codebook is able to correct a single indel or a single edit. These are the first known constructions of <inline-formula> <tex-math notation="LaTeX">\mathtt {GC} </tex-math></inline-formula>-balanced codes that correct a single indel or a single edit.