{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:GBCN6WWMJQEZHCPHYF6K7TULVF","short_pith_number":"pith:GBCN6WWM","canonical_record":{"source":{"id":"1101.1505","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.RA","submitted_at":"2011-01-07T19:16:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b625927358c82a63318ecf43f5ed06730008cc0ba41a3e3a8b3593a78d5967ba","abstract_canon_sha256":"de218f27fff6bed1dba18a983a3681e58d908340ecc45a3a5c0365d81b1b0661"},"schema_version":"1.0"},"canonical_sha256":"3044df5acc4c099389e7c17cafce8ba944c880ceca8df00d127190505d86491a","source":{"kind":"arxiv","id":"1101.1505","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.1505","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"arxiv_version","alias_value":"1101.1505v1","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.1505","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"pith_short_12","alias_value":"GBCN6WWMJQEZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"GBCN6WWMJQEZHCPH","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"GBCN6WWM","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:GBCN6WWMJQEZHCPHYF6K7TULVF","target":"record","payload":{"canonical_record":{"source":{"id":"1101.1505","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.RA","submitted_at":"2011-01-07T19:16:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b625927358c82a63318ecf43f5ed06730008cc0ba41a3e3a8b3593a78d5967ba","abstract_canon_sha256":"de218f27fff6bed1dba18a983a3681e58d908340ecc45a3a5c0365d81b1b0661"},"schema_version":"1.0"},"canonical_sha256":"3044df5acc4c099389e7c17cafce8ba944c880ceca8df00d127190505d86491a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:52.738572Z","signature_b64":"UxUpQ8OQwc1FowJcJvlIURYuPVBgchn2UyY7JBDjtENSseZ2BB0zNzV+5aKYAf0oTwsPT4K7buzp67OTE3OFCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3044df5acc4c099389e7c17cafce8ba944c880ceca8df00d127190505d86491a","last_reissued_at":"2026-05-18T04:31:52.738133Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:52.738133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.1505","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:31:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PziZpOZCASbwxTJWT3FEj68gLJvsmcz3Qq3ebnPyYVvSbcifXtzzWFNDuKdBF2eSB/yiirL/bvRcBRvFTieXBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:21:32.678056Z"},"content_sha256":"8dbde5f2a6837e305f7d0275027b2bc97222bbc948f04b3f7f98282c7dbbb943","schema_version":"1.0","event_id":"sha256:8dbde5f2a6837e305f7d0275027b2bc97222bbc948f04b3f7f98282c7dbbb943"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:GBCN6WWMJQEZHCPHYF6K7TULVF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Weight Distribution of Codes over Finite Rings","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Eimear Byrne","submitted_at":"2011-01-07T19:16:21Z","abstract_excerpt":"Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f for which the homogeneous weight distribution of C can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1505","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:31:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S4AOemp1Xw3nSy3yaExcZkUNcqnCre8P71sKHjVvA0kgYwxlnxWsRqighB5FgLdufUaRCMf2aX6vpCssJqc6Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:21:32.678420Z"},"content_sha256":"ea3c655a4e3a8173e5a39897a616ca229f495c66d05f4f32011e995d26f1677a","schema_version":"1.0","event_id":"sha256:ea3c655a4e3a8173e5a39897a616ca229f495c66d05f4f32011e995d26f1677a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GBCN6WWMJQEZHCPHYF6K7TULVF/bundle.json","state_url":"https://pith.science/pith/GBCN6WWMJQEZHCPHYF6K7TULVF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GBCN6WWMJQEZHCPHYF6K7TULVF/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":"2026-06-03T17:21:32Z","links":{"resolver":"https://pith.science/pith/GBCN6WWMJQEZHCPHYF6K7TULVF","bundle":"https://pith.science/pith/GBCN6WWMJQEZHCPHYF6K7TULVF/bundle.json","state":"https://pith.science/pith/GBCN6WWMJQEZHCPHYF6K7TULVF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GBCN6WWMJQEZHCPHYF6K7TULVF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:GBCN6WWMJQEZHCPHYF6K7TULVF","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":"de218f27fff6bed1dba18a983a3681e58d908340ecc45a3a5c0365d81b1b0661","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.RA","submitted_at":"2011-01-07T19:16:21Z","title_canon_sha256":"b625927358c82a63318ecf43f5ed06730008cc0ba41a3e3a8b3593a78d5967ba"},"schema_version":"1.0","source":{"id":"1101.1505","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.1505","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"arxiv_version","alias_value":"1101.1505v1","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.1505","created_at":"2026-05-18T04:31:52Z"},{"alias_kind":"pith_short_12","alias_value":"GBCN6WWMJQEZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"GBCN6WWMJQEZHCPH","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"GBCN6WWM","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:ea3c655a4e3a8173e5a39897a616ca229f495c66d05f4f32011e995d26f1677a","target":"graph","created_at":"2026-05-18T04:31:52Z","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":"Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f for which the homogeneous weight distribution of C can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.","authors_text":"Eimear Byrne","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.RA","submitted_at":"2011-01-07T19:16:21Z","title":"On the Weight Distribution of Codes over Finite Rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1505","kind":"arxiv","version":1},"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":"sha256:8dbde5f2a6837e305f7d0275027b2bc97222bbc948f04b3f7f98282c7dbbb943","target":"record","created_at":"2026-05-18T04:31:52Z","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":"de218f27fff6bed1dba18a983a3681e58d908340ecc45a3a5c0365d81b1b0661","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.RA","submitted_at":"2011-01-07T19:16:21Z","title_canon_sha256":"b625927358c82a63318ecf43f5ed06730008cc0ba41a3e3a8b3593a78d5967ba"},"schema_version":"1.0","source":{"id":"1101.1505","kind":"arxiv","version":1}},"canonical_sha256":"3044df5acc4c099389e7c17cafce8ba944c880ceca8df00d127190505d86491a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3044df5acc4c099389e7c17cafce8ba944c880ceca8df00d127190505d86491a","first_computed_at":"2026-05-18T04:31:52.738133Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:52.738133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UxUpQ8OQwc1FowJcJvlIURYuPVBgchn2UyY7JBDjtENSseZ2BB0zNzV+5aKYAf0oTwsPT4K7buzp67OTE3OFCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:52.738572Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.1505","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8dbde5f2a6837e305f7d0275027b2bc97222bbc948f04b3f7f98282c7dbbb943","sha256:ea3c655a4e3a8173e5a39897a616ca229f495c66d05f4f32011e995d26f1677a"],"state_sha256":"735e2220b434e59b9dfe65b7318b6024aa75afaa6ed8997578d88d91fb9370b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BglvXYBwk6w59aJtnBKuHgfXul0zxrUN80JsuYhXSbWflJieqdj2tmnq2wc3yVxdmH4gZ3deRdAWE8CqQM0xCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T17:21:32.680390Z","bundle_sha256":"df7f889353b6558ea5679e9566a618a69ce56aaec1f9bda2d0598efbde50c535"}}