LOCO Codes Can Correct as Well: Error-Correction Constrained Coding for DNA Data Storage

As a medium for cold data storage, DNA stands out as it promises significant gains in storage capacity and lifetime. However, it comes with its own data processing challenges to overcome. Constrained codes over the DNA alphabet $\{A,T,G,C\}$ have been used to design DNA sequences that are free of long homopolymers to increase stability, yet effective error detection and error correction are required to achieve reliability in data retrieval. Recently, we introduced lexicographically-ordered constrained (LOCO) codes, namely DNA LOCO (D-LOCO) codes, with error detection. In this paper, we equip our D-LOCO codes with error correction for substitution errors via syndrome-like decoding, designated as residue decoding. We only use D-LOCO codewords of indices divisible by a suitable redundancy metric $R(m) > 0$, where $m$ is the code length, for error correction. We provide the community with a construction of constrained codes forbidding runs of length higher than fixed $\ell \in \{1,2,3\}$ and $GC$-content in $\big [0.5-\frac{1}{2K},0.5+\frac{1}{2K}\big ]$ that correct $K$ segmented substitution errors, one per codeword. We call the proposed codes error-correction (EC) D-LOCO codes. We also give a list-decoding procedure with near-quadratic time-complexity in $m$ to correct double-substitution errors within EC D-LOCO codewords, which has $> 98.20\%$ average success rate. The redundancy metric is projected to require $2\log_2(m)+O(1)$-bit allocation for a length-$m$ codeword. Hence, our EC D-LOCO codes are projected to be capacity-approaching with respect to the error-free constrained system.