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 metricR(m) > 0, wheremis the code length, for error correction. The idea is that the residue, which is the index moduloR(m), of the received word index equals the residue of the index error, i.e., the index difference. We find an exhaustive list of index differences due to single-substitution errors, and we are able to recover the index of the original codeword from the residue of the received word index. This requires storing a table for index errors and their residues. Having this decoding algorithm in hand, we provide the community with a construction of constrained codes forbidding runs of length higher than fixed ℓ ∈ {1,2,3} andGC-content in [0.5−1/2K,0.5+1/2K] 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 inmto correct double-substitution errors within EC D-LOCO codewords, which has > 98.20% average success rate. The redundancy metric is projected to require 2log2(m)+O(1)-bit allocation, i.e., 2log2(m)+O(1) reduction in message bits, for a length-mcodeword. Hence, our EC D-LOCO codes are projected to be capacity-approaching with respect to the error-free constrained system.

Because of their excellent asymptotic and finite-length performance, spatially-coupled (SC) codes are a class of low-density parity-check codes that is gaining increasing attention. Multi-dimensional (MD) SC codes are constructed by connecting copies of an SC code via relocations in order to mitigate various sources of non-uniformity and improve performance in many data storage and data transmission systems. As the number of degrees of freedom in the MD-SC code design increases, appropriately exploiting them becomes more difficult because of the complexity growth of the design process. In this paper, we propose a probabilistic framework for the MD-SC code design, which is based on the gradient-descent (GD) algorithm, to design better MD codes and address this challenge. In particular, we express the expected number of short cycles, which we seek to minimize, in the graph representation of the code in terms of entries of a probability-distribution matrix that characterizes the MD-SC code design. We then find a locally-optimal probability distribution, which serves as the starting point of a finite-length algorithmic optimizer that produces the final MD-SC code. We offer the theoretical analysis as well as the algorithms, and we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower short cycle numbers compared with the available state-of-the-art. Moreover, our algorithms converge on solutions in few iterations, which confirms the complexity reduction as a result of limiting the search space via the locally-optimal GD-MD distributions.