{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OM3S5RLASBGYVCBYGW3B2LXYLP","short_pith_number":"pith:OM3S5RLA","canonical_record":{"source":{"id":"1602.05515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-17T18:06:03Z","cross_cats_sorted":[],"title_canon_sha256":"932c466c7516cc4adfbeff6f5a961e182b7996b83e8b226fb4239b651d099cdd","abstract_canon_sha256":"a23efec59701121073db14c979ff793f689decf1b5910070d005cd52c085aa5b"},"schema_version":"1.0"},"canonical_sha256":"73372ec560904d8a883835b61d2ef85bef69fd4c8613bca3e16b53227f593662","source":{"kind":"arxiv","id":"1602.05515","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05515","created_at":"2026-05-18T01:20:26Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05515v1","created_at":"2026-05-18T01:20:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05515","created_at":"2026-05-18T01:20:26Z"},{"alias_kind":"pith_short_12","alias_value":"OM3S5RLASBGY","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OM3S5RLASBGYVCBY","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OM3S5RLA","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OM3S5RLASBGYVCBYGW3B2LXYLP","target":"record","payload":{"canonical_record":{"source":{"id":"1602.05515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-17T18:06:03Z","cross_cats_sorted":[],"title_canon_sha256":"932c466c7516cc4adfbeff6f5a961e182b7996b83e8b226fb4239b651d099cdd","abstract_canon_sha256":"a23efec59701121073db14c979ff793f689decf1b5910070d005cd52c085aa5b"},"schema_version":"1.0"},"canonical_sha256":"73372ec560904d8a883835b61d2ef85bef69fd4c8613bca3e16b53227f593662","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:26.960388Z","signature_b64":"ycbF3MdJejY2x9wOirprwW2ptghPtgtnnzOsERduW184xGDP7fwogzDDTYDle2gNH+w6egRsiCgU26fvoI0oDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73372ec560904d8a883835b61d2ef85bef69fd4c8613bca3e16b53227f593662","last_reissued_at":"2026-05-18T01:20:26.959685Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:26.959685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.05515","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":"2026-05-18T01:20:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KC7fCe62BuCyGuKn6n7B322GAlJOxoP3wfwdqoG8Y2HPMFYGDtqM6IBlQck4lOwlI2YXCgV2Q47DMLv4bmtbAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:25:37.956945Z"},"content_sha256":"3401512566c14401f8efc01a09e1fcc8199bc1312032ce70859938f9ae167d14","schema_version":"1.0","event_id":"sha256:3401512566c14401f8efc01a09e1fcc8199bc1312032ce70859938f9ae167d14"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OM3S5RLASBGYVCBYGW3B2LXYLP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Endomorphisms of Cuboidal Hamming Graphs, Latin Hypercuboids of Class $r$, and Mixed MDS Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Artur Schaefer","submitted_at":"2016-02-17T18:06:03Z","abstract_excerpt":"In this paper we investigate the existence of singular endomorphisms of the cuboidal Hamming graph $H(n_1,...,n_d,S)$ over the set $\\left[ n_1\\right]\\times \\left[ n_2\\right]\\times \\cdots \\times \\left[ n_d\\right]$, where $\\left[ n\\right]=\\{1,...,n\\}$, which is a generalisation of the well-known (cubic) Hamming graph over $\\left[ n\\right]^{d}$. Two vertices in $H$ are adjacent, if their Hamming distance lies in the set $S$. In this paper $S=\\{1,...,r\\}$, for some integer $1\\leq r\\leq d-1$, and we first show that the singular endomorphisms of minimal rank ( which is the size of their image) of $H"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05515","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":"2026-05-18T01:20:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Glv19bTpu2guf6D38FvmSCZM7j/iEdoKIUm/bwT4BlpxmkMOaI/ZEE/lhM5MKcr1x8f1FqyomuWvTqIDvnzQDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:25:37.957292Z"},"content_sha256":"7f800d2317a373753c2b7f1e519cd91617aed9d1d8e64753f81ce2b53c5d6d38","schema_version":"1.0","event_id":"sha256:7f800d2317a373753c2b7f1e519cd91617aed9d1d8e64753f81ce2b53c5d6d38"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OM3S5RLASBGYVCBYGW3B2LXYLP/bundle.json","state_url":"https://pith.science/pith/OM3S5RLASBGYVCBYGW3B2LXYLP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OM3S5RLASBGYVCBYGW3B2LXYLP/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":"2026-06-27T04:25:37Z","links":{"resolver":"https://pith.science/pith/OM3S5RLASBGYVCBYGW3B2LXYLP","bundle":"https://pith.science/pith/OM3S5RLASBGYVCBYGW3B2LXYLP/bundle.json","state":"https://pith.science/pith/OM3S5RLASBGYVCBYGW3B2LXYLP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OM3S5RLASBGYVCBYGW3B2LXYLP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OM3S5RLASBGYVCBYGW3B2LXYLP","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":"a23efec59701121073db14c979ff793f689decf1b5910070d005cd52c085aa5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-17T18:06:03Z","title_canon_sha256":"932c466c7516cc4adfbeff6f5a961e182b7996b83e8b226fb4239b651d099cdd"},"schema_version":"1.0","source":{"id":"1602.05515","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05515","created_at":"2026-05-18T01:20:26Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05515v1","created_at":"2026-05-18T01:20:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05515","created_at":"2026-05-18T01:20:26Z"},{"alias_kind":"pith_short_12","alias_value":"OM3S5RLASBGY","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OM3S5RLASBGYVCBY","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OM3S5RLA","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:7f800d2317a373753c2b7f1e519cd91617aed9d1d8e64753f81ce2b53c5d6d38","target":"graph","created_at":"2026-05-18T01:20:26Z","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":"In this paper we investigate the existence of singular endomorphisms of the cuboidal Hamming graph $H(n_1,...,n_d,S)$ over the set $\\left[ n_1\\right]\\times \\left[ n_2\\right]\\times \\cdots \\times \\left[ n_d\\right]$, where $\\left[ n\\right]=\\{1,...,n\\}$, which is a generalisation of the well-known (cubic) Hamming graph over $\\left[ n\\right]^{d}$. Two vertices in $H$ are adjacent, if their Hamming distance lies in the set $S$. In this paper $S=\\{1,...,r\\}$, for some integer $1\\leq r\\leq d-1$, and we first show that the singular endomorphisms of minimal rank ( which is the size of their image) of $H","authors_text":"Artur Schaefer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-17T18:06:03Z","title":"Endomorphisms of Cuboidal Hamming Graphs, Latin Hypercuboids of Class $r$, and Mixed MDS Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05515","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":"sha256:3401512566c14401f8efc01a09e1fcc8199bc1312032ce70859938f9ae167d14","target":"record","created_at":"2026-05-18T01:20:26Z","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":"a23efec59701121073db14c979ff793f689decf1b5910070d005cd52c085aa5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-17T18:06:03Z","title_canon_sha256":"932c466c7516cc4adfbeff6f5a961e182b7996b83e8b226fb4239b651d099cdd"},"schema_version":"1.0","source":{"id":"1602.05515","kind":"arxiv","version":1}},"canonical_sha256":"73372ec560904d8a883835b61d2ef85bef69fd4c8613bca3e16b53227f593662","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73372ec560904d8a883835b61d2ef85bef69fd4c8613bca3e16b53227f593662","first_computed_at":"2026-05-18T01:20:26.959685Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:26.959685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ycbF3MdJejY2x9wOirprwW2ptghPtgtnnzOsERduW184xGDP7fwogzDDTYDle2gNH+w6egRsiCgU26fvoI0oDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:26.960388Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.05515","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3401512566c14401f8efc01a09e1fcc8199bc1312032ce70859938f9ae167d14","sha256:7f800d2317a373753c2b7f1e519cd91617aed9d1d8e64753f81ce2b53c5d6d38"],"state_sha256":"28a690463349db01842181f1b2487325e15da624dd5a8ee463ae350744c9b9a8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4J7z3+yS+sBj8h/dfxDRK9lcxnl310HYJPqAZT7lp7DVzvqwB8iAlpH9/w/6kTXbalI/V0DWaPezif8s53UJAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T04:25:37.959282Z","bundle_sha256":"8e56ac91a12a37959c89e9eaf8f2e893ae574f8175102c3e75cd4d793e527417"}}