{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:PMOCBJRRZJCCHJOZOAZLLDNBAQ","short_pith_number":"pith:PMOCBJRR","canonical_record":{"source":{"id":"1802.00148","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2018-02-01T03:48:02Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"e43ffebf9c2d95c3bb26da81d7ecf0a319df17d20cedf41461cf94a2e2d3070a","abstract_canon_sha256":"594a0cd1a3f1f2a9e7c42c641e7ec099169080ffe56cac57c03153235f857029"},"schema_version":"1.0"},"canonical_sha256":"7b1c20a631ca4423a5d97032b58da10415470c144855fa3f35901dca8ba3c95a","source":{"kind":"arxiv","id":"1802.00148","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.00148","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"arxiv_version","alias_value":"1802.00148v2","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.00148","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"pith_short_12","alias_value":"PMOCBJRRZJCC","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PMOCBJRRZJCCHJOZ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PMOCBJRR","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:PMOCBJRRZJCCHJOZOAZLLDNBAQ","target":"record","payload":{"canonical_record":{"source":{"id":"1802.00148","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2018-02-01T03:48:02Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"e43ffebf9c2d95c3bb26da81d7ecf0a319df17d20cedf41461cf94a2e2d3070a","abstract_canon_sha256":"594a0cd1a3f1f2a9e7c42c641e7ec099169080ffe56cac57c03153235f857029"},"schema_version":"1.0"},"canonical_sha256":"7b1c20a631ca4423a5d97032b58da10415470c144855fa3f35901dca8ba3c95a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:32.823386Z","signature_b64":"4FQhp4ah05DoeiLK9Qg/J/6lW8kxViBwRdQ/omcQNTAED+3AE2A4hSR6dS/h548AhFrf3THTJHBMCVB6cYAUCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b1c20a631ca4423a5d97032b58da10415470c144855fa3f35901dca8ba3c95a","last_reissued_at":"2026-05-18T00:17:32.822828Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:32.822828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.00148","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2DUsqSgBtd/A6rPqmMt2W9T6Jsq46SSbk3tQKdksJHNRwdE8k8DLlQ+UQpmTlI1xQTu7yBQKSTqDOoWTxCCTAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T08:24:48.871822Z"},"content_sha256":"56519ce62f9f27b066b09d10db96995b79502776bf6c13b4c1ed2d15f9aeb8ed","schema_version":"1.0","event_id":"sha256:56519ce62f9f27b066b09d10db96995b79502776bf6c13b4c1ed2d15f9aeb8ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:PMOCBJRRZJCCHJOZOAZLLDNBAQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"How many weights can a linear code have ?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"G\\'erard D. Cohen, Hongwei Zhu, Minjia Shi, Patrick Sol\\'e","submitted_at":"2018-02-01T03:48:02Z","abstract_excerpt":"We study the combinatorial function $L(k,q),$ the maximum number of nonzero weights a linear code of dimension $k$ over $\\F_q$ can have. We determine it completely for $q=2,$ and for $k=2,$ and provide upper and lower bounds in the general case when both $k$ and $q$ are $\\ge 3.$ A refinement $L(n,k,q),$ as well as nonlinear analogues $N(M,q)$ and $N(n,M,q),$ are also introduced and studied."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00148","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oYiWc+bqMmqDX7lQtV5uW/4mNWNtnqS+eX9dJd3fIH66HYjsMdkK+aY7FiEltpCLVzwgwQlZzRmDdifGzd2WBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T08:24:48.872532Z"},"content_sha256":"c6a50078f40530ab6f5152299411af8bf5a57f7a7a27a6e9e4ac2d1bf2700329","schema_version":"1.0","event_id":"sha256:c6a50078f40530ab6f5152299411af8bf5a57f7a7a27a6e9e4ac2d1bf2700329"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PMOCBJRRZJCCHJOZOAZLLDNBAQ/bundle.json","state_url":"https://pith.science/pith/PMOCBJRRZJCCHJOZOAZLLDNBAQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PMOCBJRRZJCCHJOZOAZLLDNBAQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-19T08:24:48Z","links":{"resolver":"https://pith.science/pith/PMOCBJRRZJCCHJOZOAZLLDNBAQ","bundle":"https://pith.science/pith/PMOCBJRRZJCCHJOZOAZLLDNBAQ/bundle.json","state":"https://pith.science/pith/PMOCBJRRZJCCHJOZOAZLLDNBAQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PMOCBJRRZJCCHJOZOAZLLDNBAQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PMOCBJRRZJCCHJOZOAZLLDNBAQ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"594a0cd1a3f1f2a9e7c42c641e7ec099169080ffe56cac57c03153235f857029","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2018-02-01T03:48:02Z","title_canon_sha256":"e43ffebf9c2d95c3bb26da81d7ecf0a319df17d20cedf41461cf94a2e2d3070a"},"schema_version":"1.0","source":{"id":"1802.00148","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.00148","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"arxiv_version","alias_value":"1802.00148v2","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.00148","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"pith_short_12","alias_value":"PMOCBJRRZJCC","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"PMOCBJRRZJCCHJOZ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"PMOCBJRR","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:c6a50078f40530ab6f5152299411af8bf5a57f7a7a27a6e9e4ac2d1bf2700329","target":"graph","created_at":"2026-05-18T00:17:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the combinatorial function $L(k,q),$ the maximum number of nonzero weights a linear code of dimension $k$ over $\\F_q$ can have. We determine it completely for $q=2,$ and for $k=2,$ and provide upper and lower bounds in the general case when both $k$ and $q$ are $\\ge 3.$ A refinement $L(n,k,q),$ as well as nonlinear analogues $N(M,q)$ and $N(n,M,q),$ are also introduced and studied.","authors_text":"G\\'erard D. Cohen, Hongwei Zhu, Minjia Shi, Patrick Sol\\'e","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2018-02-01T03:48:02Z","title":"How many weights can a linear code have ?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00148","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:56519ce62f9f27b066b09d10db96995b79502776bf6c13b4c1ed2d15f9aeb8ed","target":"record","created_at":"2026-05-18T00:17:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"594a0cd1a3f1f2a9e7c42c641e7ec099169080ffe56cac57c03153235f857029","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2018-02-01T03:48:02Z","title_canon_sha256":"e43ffebf9c2d95c3bb26da81d7ecf0a319df17d20cedf41461cf94a2e2d3070a"},"schema_version":"1.0","source":{"id":"1802.00148","kind":"arxiv","version":2}},"canonical_sha256":"7b1c20a631ca4423a5d97032b58da10415470c144855fa3f35901dca8ba3c95a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b1c20a631ca4423a5d97032b58da10415470c144855fa3f35901dca8ba3c95a","first_computed_at":"2026-05-18T00:17:32.822828Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:32.822828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4FQhp4ah05DoeiLK9Qg/J/6lW8kxViBwRdQ/omcQNTAED+3AE2A4hSR6dS/h548AhFrf3THTJHBMCVB6cYAUCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:32.823386Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.00148","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:56519ce62f9f27b066b09d10db96995b79502776bf6c13b4c1ed2d15f9aeb8ed","sha256:c6a50078f40530ab6f5152299411af8bf5a57f7a7a27a6e9e4ac2d1bf2700329"],"state_sha256":"fff25a9c1d1e6433b40a2340128a37a6e12d98ba54376425d81cbd4c4cd8ab4b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ecx9ZKUK2/sJZ8OjMSoHfPRD0wBWJINJcCEbGkCgnu1SDfU+6QibzR7wanAj40qut20q5B16Cay9MTiN9Dk5AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-19T08:24:48.875793Z","bundle_sha256":"b9257866799539067d3f3a5425639228b96e45a00d76d51e7b073b8cd53246a4"}}