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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.18323 |
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| _version_ | 1866911694090928128 |
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| author | Huang, Lei |
| author_facet | Huang, Lei |
| contents | The finite-length performance of spatially-coupled low-density parity-check (SC-LDPC) codes is strongly affected by short cycle configurations and the harmful structures induced by them. This paper studies SC-LDPC code design directly at the protograph level, where the design variables are the edge-spreading assignments specified by the partition matrix. In contrast to CLLL/Moser--Tardos based constructive frameworks for QC-SC-LDPC codes, we focus on sharper nonconstructive existence and counting bounds. By encoding cycle-activation conditions as polynomial vanishing constraints over finite grids, we apply the Combinatorial Nullstellensatz to derive sufficient memory conditions for eliminating prescribed cycle-induced harmful structures. For fully connected $(γ,κ)$ base graphs, the resulting bounds explicitly characterize the memory required to destroy all $4$-cycles as well as all $4$- and $6$-cycles, and for fixed $γ$, they are asymptotically tight up to a constant factor compared with known lower bounds. We further apply the Alon--Füredi theorem to obtain lower bounds on the number of feasible edge-spreading assignments, including an explicit counting bound for assignments that eliminate all $4$-cycles and hence yield girth at least six. These results provide a refined algebraic-combinatorial characterization of the feasible design space for high-memory SC-LDPC codes, although no corresponding construction algorithm is provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18323 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and Counting Bounds for High-Memory Spatially-Coupled Codes via the Combinatorial Nullstellensatz Huang, Lei Information Theory The finite-length performance of spatially-coupled low-density parity-check (SC-LDPC) codes is strongly affected by short cycle configurations and the harmful structures induced by them. This paper studies SC-LDPC code design directly at the protograph level, where the design variables are the edge-spreading assignments specified by the partition matrix. In contrast to CLLL/Moser--Tardos based constructive frameworks for QC-SC-LDPC codes, we focus on sharper nonconstructive existence and counting bounds. By encoding cycle-activation conditions as polynomial vanishing constraints over finite grids, we apply the Combinatorial Nullstellensatz to derive sufficient memory conditions for eliminating prescribed cycle-induced harmful structures. For fully connected $(γ,κ)$ base graphs, the resulting bounds explicitly characterize the memory required to destroy all $4$-cycles as well as all $4$- and $6$-cycles, and for fixed $γ$, they are asymptotically tight up to a constant factor compared with known lower bounds. We further apply the Alon--Füredi theorem to obtain lower bounds on the number of feasible edge-spreading assignments, including an explicit counting bound for assignments that eliminate all $4$-cycles and hence yield girth at least six. These results provide a refined algebraic-combinatorial characterization of the feasible design space for high-memory SC-LDPC codes, although no corresponding construction algorithm is provided. |
| title | Existence and Counting Bounds for High-Memory Spatially-Coupled Codes via the Combinatorial Nullstellensatz |
| topic | Information Theory |
| url | https://arxiv.org/abs/2605.18323 |