The Euclidean projection involved in the decoding of low-density parity-check (LDPC) codes with the alternating direction method of multipliers (ADMM) can be simplified by jointly using uniform quantization and look-up tables (LUTs). However, the memory requirement for the original LUT-based ADMM decoding is comparatively large. In this letter, a nonuniform quantization method is proposed to save the memory cost by minimizing the mean square error of the outputs of Euclidean projections during quantization. Simulation results over two exemplified LDPC codes show that the proposed method can achieve similar error-rate performances when compared with the original LUT-based ADMM decoding by using significantly less memory units.

Marker code is an effective coding scheme to protect data from insertions and deletions. It has potential applications in future storage systems, such as DNA storage and racetrack memory. When decoding marker codes, perfect channel state information (CSI), i.e., insertion and deletion probabilities, are required to detect insertion and deletion errors. Sometimes, the perfect CSI is not easy to obtain or the accurate channel model is unknown. Therefore, it is deserved to develop detecting algorithms for marker code without the knowledge of perfect CSI. In this paper, we propose two CSI-agnostic detecting algorithms for marker code based on deep learning. The first one is a model-driven deep learning method, which deep unfolds the original iterative detecting algorithm of marker code. In this method, CSI become weights in neural networks and these weights can be learned from training data. The second one is a data-driven method which is an end-to-end system based on the deep bidirectional gated recurrent unit network. Simulation results show that error performances of the proposed methods are significantly better than that of the original detection algorithm with CSI uncertainty. Furthermore, the proposed data-driven method exhibits better error performances than other methods for unknown channel models.

The prefixed Varshamov-Tenengolts (VT) codes, which are subsets of VT codes with predetermined prefixes, can be used for error correction over segmented edit channels. In this paper, we investigate the construction and analysis of this class of codes. First, we derive upper bounds on the size of zero-error codes for segmented edit channels with segment-by-segment decoding. Second, we establish a one-to-one correspondence between prefixed VT codes and Levenshtein codes. Based on this relation, we can obtain explicit formulas on the size of prefixed VT codes via the existing results on the size of Levenshtein codes. Third, we construct a new zero-error prefixed VT code and show that the size of the constructed code is strictly larger than that of the existing prefixed VT code for the segmented deletion channel. Finally, an efficient systematic encoding method of prefixed VT codes is proposed for the segmented edit channels.

Codes over permutations and multi-permutations have been recently proposed for efficiently storing data in flash memories. Based on the interleaving techniques for permutations, we construct two classes of permutation codes which can correct a single burst symbol-invariant deletion (SID) of length up to t. Moreover, a class of balanced multi-permutation codes capable of correcting a single burst SID of length up to t is presented. The simple decoding procedures for the constructed codes are included in proofs and verified by examples.

Permutation codes are widely studied due to their promising applications in flash memories. Based on the theory of permutation groups and subgroups, two classes of permutation codes are constructed to correct a single burst deletion of length up to a designated parameter. The proposed codes can achieve larger rates than available codes while maintaining simple interleaving structures. The decoding methods for the proposed codes are provided in proofs and verified by examples.

The alternating direction method of multipliers (ADMM) has been adapted for decoding low-density parity-check (LDPC) codes over intersymbol interference (ISI) channels in order to reduce the complexity of the traditional Turbo equalization (TE) method. However, the error rate performance of ADMM decoding may be degraded in certain ISI channels when compared with the TE at high signal-to-noise ratio regions. In order to address the problem, we derive a novel ADMM-based decoding method, called l ₂ -box ADMM decoding, for LDPC codes over ISI channels. By introducing suitable penalty functions to l ₂ -box ADMM decoding, we present an enhanced method, called l ₂ -box ADMM penalized decoding, to further improve the performance. Simulation results on several ISI channels indicate that the proposed l ₂ -box ADMM penalized decoding can achieve better error rate performances compared with the TE and the existing ADMM penalized decoding methods, especially for ISI channels with long memory, while maintaining the low-complexity advantage of the ADMM penalized decoding.