error correction system developers

Abstract: As the advent of large-scale quantum computing threatens classical encryption, Quasi-Cyclic Moderate Density Parity-Check (QC-MDPC) codes have emerged as a foundational element of Post-Quantum Cryptography (PQC). However, decoding these relatively dense codes using classical Belief Propagation algorithms often leads to failures caused by short graph cycles that stall the decoder. In cryptographic applications, these failures are severe security vulnerabilities that leak secret key information. To address this, this paper proposes a Recurrent Bipartite Graph Neural Network (R-BGNN) as proof of concept. By integrating Gated Recurrent Units (GRUs) into the message-passing framework and employing a multi-topology curriculum training strategy, the R-BGNN actively learns to break limit-cycle oscillations. Evaluated across 20 unseen matrix topologies, the proposed architecture significantly outperforms classical Bit-Flipping and Min-Sum baselines, reducing the Decoding Failure Rate (DFR) by 63.7\% at moderate noise levels ($t=10$) while converging in fewer iterations.

Abstract: Motivation: DNA data storage offers high information density and long-term stability, but accurate sequence reconstruction remains challenging under limited sequencing coverage and strong insertion, deletion, and substitution (IDS) noise. Existing reconstruction methods mainly rely on local evidence within individual read clusters. When sequencing coverage is low or indel-induced phase shifts become pronounced, such local evidence may be insufficient to distinguish closely competing candidate sequences. Results: We present Cross-Cluster Context-Guided Phase Dynamic Programming (CCP-DP), which uses dataset-level cross-cluster information to complement cluster-local evidence and selectively refines ambiguous provisional consensuses. Under the main benchmark settings, CCP-DP achieved exact reconstruction rates of 95.36%-99.98% and outperformed existing methods on multiple datasets. At $5\times$ coverage, CCP-DP improved exact reconstruction by 6.2-17.5 percentage points over the stronger of BBS and CPL across the tested IDS conditions, while maintaining low reconstruction error and computational overhead. Overall, CCP-DP introduces cross-cluster shared information as a complementary source of evidence for DNA sequence reconstruction under challenging conditions. Availability: Source code is available at https://github.com/ZixinZ3899/CCP-DP.

Abstract: For a matrix rank-metric code $\mathcal{C}$ and an integer $i$, the $i$-th subcode distance of $\mathcal{C}$ is the largest minimum rank distance of an $i$-dimensional subcode of $\mathcal{C}$. We study how subcode distances change under the Add-and-Remove construction, in which a subcode of codimension $\ell_s$ is kept and a space of dimension $\ell_a$ is added; this construction is used, with Gabidulin codes, in the $\mathsf{MIRANDA}$ signature scheme. For arbitrary matrix codes, we give inequalities between the subcode distances of the original and of the new code, and we show that Delsarte duality interchanges $\ell_s$ and $\ell_a$. For matrix Gabidulin codes, whose subcode distances are given by an explicit formula, we show that, if $\ell_s<\max\{m,n\}$, then within each block of indices $(a-1)\max\{m,n\}+1,\ldots,a\max\{m,n\}$ not exceeding $\dim_{\mathbb{F}_q}(\mathcal{C}_s)$, all but the last $\ell_s$ subcode distances are the same as for the Gabidulin code, and the remaining ones decrease by at most one; a small example shows that such a decrease can occur. We also express the minimum distance of the new code in terms of the rank distance between the added space and the common subcode. For $\mathsf{MIRANDA}$, the lower bound on the minimum distance of the dual public code used by the known distinguisher when $\ell_s=0$ is a special case of these results; for the proposed parameters, where $\ell_s>0$, the same bound holds for a subcode of codimension $\ell_s$ of the dual public code.