{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:SQBLPSVETKCUGVRGEOGTZQJVOV","short_pith_number":"pith:SQBLPSVE","canonical_record":{"source":{"id":"1101.4844","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-25T15:20:12Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"a2402344b17d119d55395f128b460a6c3d1427229380ee038628b40e906999fd","abstract_canon_sha256":"2af8e71297dab1284a99fee2d451e9213591360b66f2472257585bec335aa3c2"},"schema_version":"1.0"},"canonical_sha256":"9402b7caa49a85435626238d3cc13575635d50b444a736d1a1751027780097b7","source":{"kind":"arxiv","id":"1101.4844","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.4844","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"arxiv_version","alias_value":"1101.4844v6","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.4844","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"pith_short_12","alias_value":"SQBLPSVETKCU","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"SQBLPSVETKCUGVRG","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"SQBLPSVE","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:SQBLPSVETKCUGVRGEOGTZQJVOV","target":"record","payload":{"canonical_record":{"source":{"id":"1101.4844","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-25T15:20:12Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"a2402344b17d119d55395f128b460a6c3d1427229380ee038628b40e906999fd","abstract_canon_sha256":"2af8e71297dab1284a99fee2d451e9213591360b66f2472257585bec335aa3c2"},"schema_version":"1.0"},"canonical_sha256":"9402b7caa49a85435626238d3cc13575635d50b444a736d1a1751027780097b7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:08.736873Z","signature_b64":"ELcam45ycbV6vs6zQ4prbNZvS6J2JDV2tLEe9Dm8M6e0G7AK5ss3okw2Hc3nTpHhEmf+Y40gASqRpYjaM9fIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9402b7caa49a85435626238d3cc13575635d50b444a736d1a1751027780097b7","last_reissued_at":"2026-05-18T01:30:08.736372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:08.736372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.4844","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:30:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1pRriHVNAaHEh1N55Ql2yRUnlMDs8eLoIXjWJakBSmX++Vp2WyX59aK9q6VDjrObstp9jhNax2kcYM5cGaAPCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:44:30.385250Z"},"content_sha256":"0189ea37623d3238642aeb7147e89ab244a98dd7b1ef5d68f974c397a75699cb","schema_version":"1.0","event_id":"sha256:0189ea37623d3238642aeb7147e89ab244a98dd7b1ef5d68f974c397a75699cb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:SQBLPSVETKCUGVRGEOGTZQJVOV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Properties of Codes with Two Homogeneous Weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CO","authors_text":"Alison Sneyd, Eimear Byrne, Michael Kiermaier","submitted_at":"2011-01-25T15:20:12Z","abstract_excerpt":"Delsarte showed that for any projective linear code over a finite field of characteristic p with two nonzero Hamming weights w1 < w2 there exist positive integers u and s such that w1 = (p^s)u and w2 = (p^s)(u+1). Moreover, he showed that the additive group of such a code has a strongly regular Cayley graph. Here we show that for any proper regular projective linear code C over a finite Frobenius ring with two integral nonzero homogeneous weights w1 < w2, there is a positive integer d, a divisor of the order of C, and positive integer u such that w1 = du and w2 = d(u+1). In doing so, we give a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4844","kind":"arxiv","version":6},"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":"2026-05-18T01:30:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FM/Al/EhOEb01vPEgJtqfaB0SmqQxFSp69XWN30qYyL9NfT4lP2zaby2c3qiKAlGaFIy815053x1I7+Ihlb1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:44:30.385624Z"},"content_sha256":"5d7eedca0c8186ce5f426f7f927a72eab1e549fa0798d48a121df70919a8515b","schema_version":"1.0","event_id":"sha256:5d7eedca0c8186ce5f426f7f927a72eab1e549fa0798d48a121df70919a8515b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SQBLPSVETKCUGVRGEOGTZQJVOV/bundle.json","state_url":"https://pith.science/pith/SQBLPSVETKCUGVRGEOGTZQJVOV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SQBLPSVETKCUGVRGEOGTZQJVOV/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":"2026-05-27T20:44:30Z","links":{"resolver":"https://pith.science/pith/SQBLPSVETKCUGVRGEOGTZQJVOV","bundle":"https://pith.science/pith/SQBLPSVETKCUGVRGEOGTZQJVOV/bundle.json","state":"https://pith.science/pith/SQBLPSVETKCUGVRGEOGTZQJVOV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SQBLPSVETKCUGVRGEOGTZQJVOV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:SQBLPSVETKCUGVRGEOGTZQJVOV","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":"2af8e71297dab1284a99fee2d451e9213591360b66f2472257585bec335aa3c2","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-25T15:20:12Z","title_canon_sha256":"a2402344b17d119d55395f128b460a6c3d1427229380ee038628b40e906999fd"},"schema_version":"1.0","source":{"id":"1101.4844","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.4844","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"arxiv_version","alias_value":"1101.4844v6","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.4844","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"pith_short_12","alias_value":"SQBLPSVETKCU","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"SQBLPSVETKCUGVRG","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"SQBLPSVE","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:5d7eedca0c8186ce5f426f7f927a72eab1e549fa0798d48a121df70919a8515b","target":"graph","created_at":"2026-05-18T01:30:08Z","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":"Delsarte showed that for any projective linear code over a finite field of characteristic p with two nonzero Hamming weights w1 < w2 there exist positive integers u and s such that w1 = (p^s)u and w2 = (p^s)(u+1). Moreover, he showed that the additive group of such a code has a strongly regular Cayley graph. Here we show that for any proper regular projective linear code C over a finite Frobenius ring with two integral nonzero homogeneous weights w1 < w2, there is a positive integer d, a divisor of the order of C, and positive integer u such that w1 = du and w2 = d(u+1). In doing so, we give a","authors_text":"Alison Sneyd, Eimear Byrne, Michael Kiermaier","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-25T15:20:12Z","title":"Properties of Codes with Two Homogeneous Weights"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4844","kind":"arxiv","version":6},"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":"sha256:0189ea37623d3238642aeb7147e89ab244a98dd7b1ef5d68f974c397a75699cb","target":"record","created_at":"2026-05-18T01:30:08Z","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":"2af8e71297dab1284a99fee2d451e9213591360b66f2472257585bec335aa3c2","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-25T15:20:12Z","title_canon_sha256":"a2402344b17d119d55395f128b460a6c3d1427229380ee038628b40e906999fd"},"schema_version":"1.0","source":{"id":"1101.4844","kind":"arxiv","version":6}},"canonical_sha256":"9402b7caa49a85435626238d3cc13575635d50b444a736d1a1751027780097b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9402b7caa49a85435626238d3cc13575635d50b444a736d1a1751027780097b7","first_computed_at":"2026-05-18T01:30:08.736372Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:08.736372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ELcam45ycbV6vs6zQ4prbNZvS6J2JDV2tLEe9Dm8M6e0G7AK5ss3okw2Hc3nTpHhEmf+Y40gASqRpYjaM9fIAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:08.736873Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.4844","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0189ea37623d3238642aeb7147e89ab244a98dd7b1ef5d68f974c397a75699cb","sha256:5d7eedca0c8186ce5f426f7f927a72eab1e549fa0798d48a121df70919a8515b"],"state_sha256":"6f5ef2d76b04ed6514a5cc1a6849cc0d07c443a1efe0869af2f2924ad45198a9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1R0lP/JYGKtLww/+kcmh4wrADz2bM5gldtVmKPUqQd7Y9a0z0/Kz7uCQNO6OJkWW68IQJk/l76T9NSLm55DABw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T20:44:30.387736Z","bundle_sha256":"a96f4c6a73dd0ba806a54de6c35e6425731ed5de0a42098e9b364ec68b333966"}}