Ke Liu

Chenyuan Jia, Qingqing Peng, Ke Liu et al.

The minimum distance problem (MDP) for low-density parity-check (LDPC) codes is a central problem in coding theory and is closely related to the analysis of low-weight codewords and error-floor behavior. Although the unrestricted MDP is computationally intractable, its complexity under degree constraints that commonly occur in LDPC code design has remained less clear. In this paper, we study the MDP for left regular and biregular Tanner graphs. We prove that the problem is $\mathrm{NP}$-complete and $\mathrm{W}[1]$-complete for $J$-left regular Tanner graphs for every fixed $J\geq 3$, and also for $(3,3)$-regular bipartite graphs. We further establish $\mathrm{W}[1]$-completeness for $(J,K)$-regular instances for every fixed $J,K\geq 3$. The reductions are based on a degree-preserving transformation framework consisting of hyperedge decomposition, check node splitting, and controlled variable replication. These transformations transfer hardness between different degree distributions while preserving explicit bijections among nonzero codewords, even covers, and nonempty $(a,0)$-trapping sets. The results delineate the computational limits of exact LDPC code analysis under natural regularity constraints.

Xiaoliang Wu, Chongxin Gan, Ke Liu et al.

Deep neural network-based automatic speaker verification (ASV) systems achieve impressive performance but their embedding representations remain opaque, lacking a structured and perceptually verifiable explanation of the vocal characteristics they encode. Existing approaches either require annotation of speaker attributes or introduce alternative representations whose interpretability is unvalidated with listeners. We propose Listenable Interpretable Speaker Embeddings (LISE), a label-free framework that decomposes pretrained speaker embeddings into a small set of components. This decomposition yields a structured representation that supports the analysis of what information has been encoded by speaker embeddings. LISE preserves ASV performance with negligible EER degradation on x-vector and ECAPA-TDNN. Crucially, the interpretability of these components for human listeners is demonstrated through listening experiments, where participants distinguished speakers with 83.9% accuracy.