Correcting a Single Indel/Edit for DNA-Based Data Storage: Linear-Time Encoders and Order-Optimality - Cai, Kui; Chee, Yeow Meng; Gabrys, Ryan; Kiah, Han Mao; Nguyen, Tuan Thanh (IEEE transactions on information theory, 2021)
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 \lceil {\log \text {n}}\rceil+\text {O}(\log \log \text {n}) redundancy bits, while the other corrects a single indel with \lceil {\log \text {n}}\rceil+2 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 \mathtt {GC} -balanced constraint and require that exactly half of the symbols of any DNA codeword to be either \mathtt {C} or \mathtt {G} . In particular, via a modification of Knuth's balancing technique, we provide a linear-time map that translates binary messages into \mathtt {GC} -balanced codewords and the resulting codebook is able to correct a single indel or a single edit. These are the first known constructions of \mathtt {GC} -balanced codes that correct a single indel or a single edit.