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The asymptotic tradeoff among block length $N$, code rate $R$, and error probability $P$ is analyzed.\n  Given a tradeoff between $N,P$ and a tradeoff between $N,R$, we return an interpolating tradeoff among $N,R,P$ (Theorem 5). $\\def\\Capacity{\\text{Capacity}}$Quantitatively, if $P=\\exp(-N^{\\beta^*})$ is possible for some $\\beta^*$ and if $R=\\Capacity-N^{1/\\mu^*}$ is possible for some $1/\\mu^*$, then $(P,R)=(\\exp(-N^{\\beta'}),\\Capacity-N^{-1/\\mu'})$ is possible for some pair $(\\beta',1/\\mu'"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.08112","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2018-12-19T17:42:44Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"ce2caca83602ac04c3a06bbbc3e82cfe085c710e88ddaa9de48cab2fecddc7c7","abstract_canon_sha256":"10e996bc1c207cc25904c635b8d93bed7574c18e8edc672677a2487a3c93d021"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:54.810940Z","signature_b64":"nnUc5wztnvy4nhX9lKGDaSBib4repJ2P9afPJU2o2XcnG+XWUWHREmn+ebiLKX96dzgZ9fGmAQefivCGaQf1Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70aba7a5c657be1926c15a24e087b28513db626126c71d5d9998309c0603bbfe","last_reissued_at":"2026-05-17T23:57:54.810387Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:54.810387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polar-like Codes and Asymptotic Tradeoff among Block Length, Code Rate, and Error Probability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Hsin-Po Wang, Iwan Duursma","submitted_at":"2018-12-19T17:42:44Z","abstract_excerpt":"A general framework is proposed that includes polar codes over arbitrary channels with arbitrary kernels. The asymptotic tradeoff among block length $N$, code rate $R$, and error probability $P$ is analyzed.\n  Given a tradeoff between $N,P$ and a tradeoff between $N,R$, we return an interpolating tradeoff among $N,R,P$ (Theorem 5). $\\def\\Capacity{\\text{Capacity}}$Quantitatively, if $P=\\exp(-N^{\\beta^*})$ is possible for some $\\beta^*$ and if $R=\\Capacity-N^{1/\\mu^*}$ is possible for some $1/\\mu^*$, then $(P,R)=(\\exp(-N^{\\beta'}),\\Capacity-N^{-1/\\mu'})$ is possible for some pair $(\\beta',1/\\mu'"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08112","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1812.08112","created_at":"2026-05-17T23:57:54.810475+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.08112v1","created_at":"2026-05-17T23:57:54.810475+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.08112","created_at":"2026-05-17T23:57:54.810475+00:00"},{"alias_kind":"pith_short_12","alias_value":"OCV2PJOGK67B","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"OCV2PJOGK67BSJWB","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"OCV2PJOG","created_at":"2026-05-18T12:32:43.782077+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU","json":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU.json","graph_json":"https://pith.science/api/pith-number/OCV2PJOGK67BSJWBLISOBB5SQU/graph.json","events_json":"https://pith.science/api/pith-number/OCV2PJOGK67BSJWBLISOBB5SQU/events.json","paper":"https://pith.science/paper/OCV2PJOG"},"agent_actions":{"view_html":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU","download_json":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU.json","view_paper":"https://pith.science/paper/OCV2PJOG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.08112&json=true","fetch_graph":"https://pith.science/api/pith-number/OCV2PJOGK67BSJWBLISOBB5SQU/graph.json","fetch_events":"https://pith.science/api/pith-number/OCV2PJOGK67BSJWBLISOBB5SQU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU/action/storage_attestation","attest_author":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU/action/author_attestation","sign_citation":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU/action/citation_signature","submit_replication":"https://pith.science/pith/OCV2PJOGK67BSJWBLISOBB5SQU/action/replication_record"}},"created_at":"2026-05-17T23:57:54.810475+00:00","updated_at":"2026-05-17T23:57:54.810475+00:00"}