{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:ULNTMQAEVXHJUPT76QZX3MLP2Q","short_pith_number":"pith:ULNTMQAE","canonical_record":{"source":{"id":"2010.12429","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-10-23T14:21:43Z","cross_cats_sorted":["math.IT","math.RA"],"title_canon_sha256":"4198a91abaacf2143cb60c4c4a32dcdc7646f5524811b16e7769f449121d1426","abstract_canon_sha256":"76d57a8c372d775a2cbd30a5b804bc12c205694de6378d5f60c5b92d2614e463"},"schema_version":"1.0"},"canonical_sha256":"a2db364004adce9a3e7ff4337db16fd424cc03a044dc4ef64137a41897297217","source":{"kind":"arxiv","id":"2010.12429","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.12429","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"arxiv_version","alias_value":"2010.12429v1","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.12429","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"pith_short_12","alias_value":"ULNTMQAEVXHJ","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"pith_short_16","alias_value":"ULNTMQAEVXHJUPT7","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"pith_short_8","alias_value":"ULNTMQAE","created_at":"2026-07-05T01:45:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:ULNTMQAEVXHJUPT76QZX3MLP2Q","target":"record","payload":{"canonical_record":{"source":{"id":"2010.12429","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-10-23T14:21:43Z","cross_cats_sorted":["math.IT","math.RA"],"title_canon_sha256":"4198a91abaacf2143cb60c4c4a32dcdc7646f5524811b16e7769f449121d1426","abstract_canon_sha256":"76d57a8c372d775a2cbd30a5b804bc12c205694de6378d5f60c5b92d2614e463"},"schema_version":"1.0"},"canonical_sha256":"a2db364004adce9a3e7ff4337db16fd424cc03a044dc4ef64137a41897297217","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:45:37.189463Z","signature_b64":"sFwwXv8EoX2ChsEllRM8j5boA7u7Q63KmkfWNcnxdJ/dxo5kvJ+chsBHQshbDdK34VNg+/x4jUfvC3g3TNvHCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2db364004adce9a3e7ff4337db16fd424cc03a044dc4ef64137a41897297217","last_reissued_at":"2026-07-05T01:45:37.188979Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:45:37.188979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2010.12429","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":"2026-07-05T01:45:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sCozOFfA6Dbkv2K4Pu0GV3ky99BYzqhP0WB6panM07ZOBNUbwbxpqlk6tL3Zxmn8HsWFb4voLtt2/1DaMkUCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T13:51:23.937696Z"},"content_sha256":"b7716029ab6a0cad8e9a042ad9215570c8755b9bf5d4e3c6e0a5a8c963d0c5f0","schema_version":"1.0","event_id":"sha256:b7716029ab6a0cad8e9a042ad9215570c8755b9bf5d4e3c6e0a5a8c963d0c5f0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:ULNTMQAEVXHJUPT76QZX3MLP2Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Relative projective group codes over chain rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.RA"],"primary_cat":"cs.IT","authors_text":"Sihuang Hu, Simon Eisenbarth","submitted_at":"2020-10-23T14:21:43Z","abstract_excerpt":"A structure theorem of the group codes which are relative projective for the subgroup $\\lbrace 1 \\rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $\\ell$ in the group algebra $\\mathbb{F}G$, where $\\mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.12429","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2010.12429/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:45:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4H2TsAOB2e94DS2+6Fk6ETVjmOtnFabtCBjm1AeEn+Kev4ghdjU12IdEeX4RnipV9K5ibXvtw8jBfRfgvEDuDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T13:51:23.938061Z"},"content_sha256":"e9166324b6e0822edc7bccc649ae6229c9b015065c078bffd52b7acecd17a253","schema_version":"1.0","event_id":"sha256:e9166324b6e0822edc7bccc649ae6229c9b015065c078bffd52b7acecd17a253"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ULNTMQAEVXHJUPT76QZX3MLP2Q/bundle.json","state_url":"https://pith.science/pith/ULNTMQAEVXHJUPT76QZX3MLP2Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ULNTMQAEVXHJUPT76QZX3MLP2Q/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":"2026-07-06T13:51:23Z","links":{"resolver":"https://pith.science/pith/ULNTMQAEVXHJUPT76QZX3MLP2Q","bundle":"https://pith.science/pith/ULNTMQAEVXHJUPT76QZX3MLP2Q/bundle.json","state":"https://pith.science/pith/ULNTMQAEVXHJUPT76QZX3MLP2Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ULNTMQAEVXHJUPT76QZX3MLP2Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:ULNTMQAEVXHJUPT76QZX3MLP2Q","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":"76d57a8c372d775a2cbd30a5b804bc12c205694de6378d5f60c5b92d2614e463","cross_cats_sorted":["math.IT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-10-23T14:21:43Z","title_canon_sha256":"4198a91abaacf2143cb60c4c4a32dcdc7646f5524811b16e7769f449121d1426"},"schema_version":"1.0","source":{"id":"2010.12429","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.12429","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"arxiv_version","alias_value":"2010.12429v1","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.12429","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"pith_short_12","alias_value":"ULNTMQAEVXHJ","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"pith_short_16","alias_value":"ULNTMQAEVXHJUPT7","created_at":"2026-07-05T01:45:37Z"},{"alias_kind":"pith_short_8","alias_value":"ULNTMQAE","created_at":"2026-07-05T01:45:37Z"}],"graph_snapshots":[{"event_id":"sha256:e9166324b6e0822edc7bccc649ae6229c9b015065c078bffd52b7acecd17a253","target":"graph","created_at":"2026-07-05T01:45:37Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2010.12429/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A structure theorem of the group codes which are relative projective for the subgroup $\\lbrace 1 \\rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $\\ell$ in the group algebra $\\mathbb{F}G$, where $\\mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight.","authors_text":"Sihuang Hu, Simon Eisenbarth","cross_cats":["math.IT","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-10-23T14:21:43Z","title":"Relative projective group codes over chain rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.12429","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:b7716029ab6a0cad8e9a042ad9215570c8755b9bf5d4e3c6e0a5a8c963d0c5f0","target":"record","created_at":"2026-07-05T01:45:37Z","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":"76d57a8c372d775a2cbd30a5b804bc12c205694de6378d5f60c5b92d2614e463","cross_cats_sorted":["math.IT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2020-10-23T14:21:43Z","title_canon_sha256":"4198a91abaacf2143cb60c4c4a32dcdc7646f5524811b16e7769f449121d1426"},"schema_version":"1.0","source":{"id":"2010.12429","kind":"arxiv","version":1}},"canonical_sha256":"a2db364004adce9a3e7ff4337db16fd424cc03a044dc4ef64137a41897297217","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a2db364004adce9a3e7ff4337db16fd424cc03a044dc4ef64137a41897297217","first_computed_at":"2026-07-05T01:45:37.188979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:45:37.188979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sFwwXv8EoX2ChsEllRM8j5boA7u7Q63KmkfWNcnxdJ/dxo5kvJ+chsBHQshbDdK34VNg+/x4jUfvC3g3TNvHCA==","signature_status":"signed_v1","signed_at":"2026-07-05T01:45:37.189463Z","signed_message":"canonical_sha256_bytes"},"source_id":"2010.12429","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b7716029ab6a0cad8e9a042ad9215570c8755b9bf5d4e3c6e0a5a8c963d0c5f0","sha256:e9166324b6e0822edc7bccc649ae6229c9b015065c078bffd52b7acecd17a253"],"state_sha256":"28d2e00be39aef640f1bbf4e7af98800bc37e4ac8049727bd4be3995511df6e2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xqS3li2R0QAVOc3vTtn8QAtwjR/9O7cMjc/1SZHxgcn9hOMR1jprWPeds2KXx03PX8Ccx716i+qZHpGptqlUDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T13:51:23.939955Z","bundle_sha256":"3b23cb07bed94e3a4c75652c3554a7a47aa52ac5eff3d7448693205173bdaf7c"}}