{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:BBANFHPKVBYPLDF43FPDN3CIPQ","short_pith_number":"pith:BBANFHPK","canonical_record":{"source":{"id":"1904.05227","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-10T14:52:13Z","cross_cats_sorted":["math.AC","math.IT"],"title_canon_sha256":"df722660015cb72171f5049fed876780540a86ae692a8294d228d0c850f5b0f8","abstract_canon_sha256":"21a5136e27269756d578426a41a630811df105336bb395a5dfcb1a5132891445"},"schema_version":"1.0"},"canonical_sha256":"0840d29deaa870f58cbcd95e36ec487c0eaf67fc6d89e96ae4882675d18eb78b","source":{"kind":"arxiv","id":"1904.05227","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.05227","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"arxiv_version","alias_value":"1904.05227v1","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.05227","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"pith_short_12","alias_value":"BBANFHPKVBYP","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"BBANFHPKVBYPLDF4","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"BBANFHPK","created_at":"2026-05-18T12:33:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:BBANFHPKVBYPLDF43FPDN3CIPQ","target":"record","payload":{"canonical_record":{"source":{"id":"1904.05227","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-10T14:52:13Z","cross_cats_sorted":["math.AC","math.IT"],"title_canon_sha256":"df722660015cb72171f5049fed876780540a86ae692a8294d228d0c850f5b0f8","abstract_canon_sha256":"21a5136e27269756d578426a41a630811df105336bb395a5dfcb1a5132891445"},"schema_version":"1.0"},"canonical_sha256":"0840d29deaa870f58cbcd95e36ec487c0eaf67fc6d89e96ae4882675d18eb78b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:53.758378Z","signature_b64":"5KtTVvAHf3/PGifPMF4nVopVZCvRWf0abQOEKAZXk2Ot1PJxJKlz5PZBjQSskFrlm6/IbuQPB7Cwqx9lyU1VAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0840d29deaa870f58cbcd95e36ec487c0eaf67fc6d89e96ae4882675d18eb78b","last_reissued_at":"2026-05-17T23:48:53.757809Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:53.757809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.05227","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":"2026-05-17T23:48:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ioKkjIYzj9wfBFLQZImayIAHoV0/+R9rha3yGeDwpzz8XeCD20TjUAIAZs9OTwOWKvkWNGHsC/HeDulJHR2GDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:21:41.769766Z"},"content_sha256":"3e3266501857292e2b3fa1d54ebc9c7bd4685d844a9e5e71239de09f7980106e","schema_version":"1.0","event_id":"sha256:3e3266501857292e2b3fa1d54ebc9c7bd4685d844a9e5e71239de09f7980106e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:BBANFHPKVBYPLDF43FPDN3CIPQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tensor Representation of Rank-Metric Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.IT"],"primary_cat":"cs.IT","authors_text":"Alberto Ravagnani, Alessandro Neri, Eimear Byrne, John Sheekey","submitted_at":"2019-04-10T14:52:13Z","abstract_excerpt":"We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor rank of a code to be the tensor rank of its generating tensors, and propose that this quantity is a significant coding theoretic parameter. By a result on the tensor rank of Kruskal from the 1970s, the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k+d-1$. We call codes that meet this bound minimal t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.05227","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":"2026-05-17T23:48:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/nPXXHVwuE8aNZVSJ8VFVRRmvRXT2TBa9vKBT27PL0LrrJL17+1vmUJnaeLia+dLpQA8VjqivpWe6h5sWtkcDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:21:41.770114Z"},"content_sha256":"a06dba0443a51fd67f5935d890c25bc6dadd28821082f09b1936ba542a794711","schema_version":"1.0","event_id":"sha256:a06dba0443a51fd67f5935d890c25bc6dadd28821082f09b1936ba542a794711"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BBANFHPKVBYPLDF43FPDN3CIPQ/bundle.json","state_url":"https://pith.science/pith/BBANFHPKVBYPLDF43FPDN3CIPQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BBANFHPKVBYPLDF43FPDN3CIPQ/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":"2026-06-03T05:21:41Z","links":{"resolver":"https://pith.science/pith/BBANFHPKVBYPLDF43FPDN3CIPQ","bundle":"https://pith.science/pith/BBANFHPKVBYPLDF43FPDN3CIPQ/bundle.json","state":"https://pith.science/pith/BBANFHPKVBYPLDF43FPDN3CIPQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BBANFHPKVBYPLDF43FPDN3CIPQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:BBANFHPKVBYPLDF43FPDN3CIPQ","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":"21a5136e27269756d578426a41a630811df105336bb395a5dfcb1a5132891445","cross_cats_sorted":["math.AC","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-10T14:52:13Z","title_canon_sha256":"df722660015cb72171f5049fed876780540a86ae692a8294d228d0c850f5b0f8"},"schema_version":"1.0","source":{"id":"1904.05227","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.05227","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"arxiv_version","alias_value":"1904.05227v1","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.05227","created_at":"2026-05-17T23:48:53Z"},{"alias_kind":"pith_short_12","alias_value":"BBANFHPKVBYP","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"BBANFHPKVBYPLDF4","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"BBANFHPK","created_at":"2026-05-18T12:33:12Z"}],"graph_snapshots":[{"event_id":"sha256:a06dba0443a51fd67f5935d890c25bc6dadd28821082f09b1936ba542a794711","target":"graph","created_at":"2026-05-17T23:48:53Z","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 present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor rank of a code to be the tensor rank of its generating tensors, and propose that this quantity is a significant coding theoretic parameter. By a result on the tensor rank of Kruskal from the 1970s, the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k+d-1$. We call codes that meet this bound minimal t","authors_text":"Alberto Ravagnani, Alessandro Neri, Eimear Byrne, John Sheekey","cross_cats":["math.AC","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-10T14:52:13Z","title":"Tensor Representation of Rank-Metric Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.05227","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":"sha256:3e3266501857292e2b3fa1d54ebc9c7bd4685d844a9e5e71239de09f7980106e","target":"record","created_at":"2026-05-17T23:48:53Z","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":"21a5136e27269756d578426a41a630811df105336bb395a5dfcb1a5132891445","cross_cats_sorted":["math.AC","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-10T14:52:13Z","title_canon_sha256":"df722660015cb72171f5049fed876780540a86ae692a8294d228d0c850f5b0f8"},"schema_version":"1.0","source":{"id":"1904.05227","kind":"arxiv","version":1}},"canonical_sha256":"0840d29deaa870f58cbcd95e36ec487c0eaf67fc6d89e96ae4882675d18eb78b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0840d29deaa870f58cbcd95e36ec487c0eaf67fc6d89e96ae4882675d18eb78b","first_computed_at":"2026-05-17T23:48:53.757809Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:53.757809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5KtTVvAHf3/PGifPMF4nVopVZCvRWf0abQOEKAZXk2Ot1PJxJKlz5PZBjQSskFrlm6/IbuQPB7Cwqx9lyU1VAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:53.758378Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.05227","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e3266501857292e2b3fa1d54ebc9c7bd4685d844a9e5e71239de09f7980106e","sha256:a06dba0443a51fd67f5935d890c25bc6dadd28821082f09b1936ba542a794711"],"state_sha256":"e76f65ccd4918d3b7d7e9a253b4e6c6da6e014e6b145252c987bfa704ff7019d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y5oQ9lIakCwJz2PZ7lchQE7jvi5WYCRAFzoPfy4HuZtbr8mItUCScVImDNbSrO1ZrCI2lCfHMEzifPahxDc2CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T05:21:41.772088Z","bundle_sha256":"dd5462f2ec8a161e09e083ab6f40e15e01ce8d9faea9d5d199636b0a96172423"}}