Improved constructions of secondary structure avoidance codes for DNA sequences

In a DNA sequence, we have the celebrated Watson-Crick complement $\overline{T}=A, \overline{A}= {T}, \overline{C}= G$, and $\overline{G}= C$. The phenomenon of secondary structure refers to the tendency of a single stranded DNA sequence to fold back upon itself, and it is usually caused by the existence of two non-overlapping reverse complement consecutive subsequences, defined as follows. Given an integer $m \geq 2$, two non-overlapping consecutive subsequences of length m, denoted as $x=(x_{1}, x_{m})$ and $y=(y_{1}, y_{m})$, are called reverse complement if $x_{i}=\overline{y_{m-i+1}}$ for 1$\leq i \leq m$. The property of secondary structure avoidance (SSA) forbids a sequence to contain such reverse complement subsequences, and it is a key criterion in the design of single-stranded DNA sequences for DNA computing and storage. In this paper, we improve on a recent result of Nguyen et al., by introducing explicit constructions of secondary structure avoidance codes and analyzing the capacity for any given m. In particular, our constructions have optimal rate 1.1679bits/nt and 1.5515bits/nt when m = 2 and m + 3, respectively.