Efficient Explicit and Pseudo-Random Constructions of Constrained Codes for DNA Storage

GC-content and homopolymers are two constraints widely considered in DNA storage. Extensive experiments show that DNA strings with too high/low GC-content and/or long homopolymers are more likely to suffer from errors. We say a length-n DNA string satisfies the $(\epsilon ,\ell)$ -constraints if no homopolymer exceeds $\ell $ and its GC-content falls between $(0.5-\epsilon)n$ and $(0.5+\epsilon)n$ , where $\epsilon $ is a positive real number and $\ell $ is a positive integer; say it satisfies the $(\delta ,\ell)$ -prefix-constraints if no homopolymer exceeds $\ell $ and the GC-content of its length-i prefix falls between $i/2-\delta $ and $i/2+\delta $ for any $i \in \{1, 2, \ldots , n\}$ , where $\delta $ is a positive integer. In this paper, we realize the first enumerative coding for the capacity-achieving $(\epsilon ,\ell)$ -constrained codes and $(\delta ,\ell)$ -prefix-constrained codes with polynomial encoding/decoding complexity. Meanwhile, we propose an efficient pseudo-random construction of capacity-approaching $(\epsilon ,\ell)$ -constrained codes and $(\epsilon ,\ell)$ -constrained error correction codes. We further generalize the idea and establish a systematic framework of pseudo-random construction for a wide range of codes.