{"paper":{"title":"Existence and Counting Bounds for High-Memory Spatially-Coupled Codes via the Combinatorial Nullstellensatz","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Applies Combinatorial Nullstellensatz to derive sufficient memory conditions for SC-LDPC protographs to destroy all 4-cycles and all 4- and 6-cycles, plus lower bounds on the number of such feasible edge-spreading assignments.","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Lei Huang","submitted_at":"2026-05-18T12:41:53Z","abstract_excerpt":"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 gr"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Cycle-activation conditions for the harmful structures can be encoded as polynomial vanishing constraints over finite grids in a way that permits direct application of the Combinatorial Nullstellensatz to obtain the stated memory thresholds.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Applies Combinatorial Nullstellensatz to derive sufficient memory conditions for SC-LDPC protographs to destroy all 4-cycles and all 4- and 6-cycles, plus lower bounds on the number of such feasible edge-spreading assignments.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"316199670de72d82fba5eb13dd34ca241ccbe62b76c01e28905b72b7908cf1b8"},"source":{"id":"2605.18323","kind":"arxiv","version":1},"verdict":{"id":"8516a501-fc73-4f55-8c2c-f3d7bb0074ea","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:58:53.585156Z","strongest_claim":"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.","one_line_summary":"Applies Combinatorial Nullstellensatz to derive sufficient memory conditions for SC-LDPC protographs to destroy all 4-cycles and all 4- and 6-cycles, plus lower bounds on the number of such feasible edge-spreading assignments.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Cycle-activation conditions for the harmful structures can be encoded as polynomial vanishing constraints over finite grids in a way that permits direct application of the Combinatorial Nullstellensatz to obtain the stated memory thresholds.","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18323/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.185480Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T23:21:58.859050Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e5fca57ddc7fe2f0e6003f705116e9b2c9b244c516ec7bb7ffbdca2ba6c219d8"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}