{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:N2Y3MGNKVUHBHTQEPHBFNWADXG","short_pith_number":"pith:N2Y3MGNK","canonical_record":{"source":{"id":"1610.06432","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T14:33:40Z","cross_cats_sorted":[],"title_canon_sha256":"a066d1549e345d0973f312e799acb3d1c3481bd3d8fecf175ba630c083a6f893","abstract_canon_sha256":"9bf406264e7f612dd42f0fcdb5c5677a3e5d93f09386329e94c59ac5c908de95"},"schema_version":"1.0"},"canonical_sha256":"6eb1b619aaad0e13ce0479c256d803b99461f67e02cd34881bf36f5020f99d15","source":{"kind":"arxiv","id":"1610.06432","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06432","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06432v2","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06432","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"N2Y3MGNKVUHB","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"N2Y3MGNKVUHBHTQE","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"N2Y3MGNK","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:N2Y3MGNKVUHBHTQEPHBFNWADXG","target":"record","payload":{"canonical_record":{"source":{"id":"1610.06432","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T14:33:40Z","cross_cats_sorted":[],"title_canon_sha256":"a066d1549e345d0973f312e799acb3d1c3481bd3d8fecf175ba630c083a6f893","abstract_canon_sha256":"9bf406264e7f612dd42f0fcdb5c5677a3e5d93f09386329e94c59ac5c908de95"},"schema_version":"1.0"},"canonical_sha256":"6eb1b619aaad0e13ce0479c256d803b99461f67e02cd34881bf36f5020f99d15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:58.532170Z","signature_b64":"r60snjyWl9WutLyb4v/YDWSpIya/psKR49TVsuzATOx+ZZauUkaS5YeHPMurvm8K1kQsDQgvRWBcFIqlvCXxAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6eb1b619aaad0e13ce0479c256d803b99461f67e02cd34881bf36f5020f99d15","last_reissued_at":"2026-05-18T00:45:58.531702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:58.531702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.06432","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FYW4CrU5RaKVR/QNzH/HWhDpKuGSrxw2sY5hmBrwbemnuuai2A517K5lQ3kKpxoo8lCTmmXFwgmMOc/3HKfKAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:10:25.470279Z"},"content_sha256":"65b4d5d8c0358c4c23e3510fa0bfe107b8ac1c643bd3366316e6b90df5513d3c","schema_version":"1.0","event_id":"sha256:65b4d5d8c0358c4c23e3510fa0bfe107b8ac1c643bd3366316e6b90df5513d3c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:N2Y3MGNKVUHBHTQEPHBFNWADXG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some Nonexistence Results for Strong External Difference Families Using Character Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Douglas R. Stinson, William J. Martin","submitted_at":"2016-10-20T14:33:40Z","abstract_excerpt":"In this paper, we study the existence of $(v,m,k,\\lambda)$-strong external difference families (SEDFs). We use character-theoretic techniques to show that no SEDF exists when $v$ is prime, $k > 1$ and $m > 2$. In the case where $v$ is the product of two distinct odd primes, some necessary conditions are derived, which can be used to rule out certain parameter sets. Further, we show that, when $m=3$ or $4$ and $v > m$, a $(v,m,k,\\lambda)$-SEDF does not exist."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06432","kind":"arxiv","version":2},"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":"2026-05-18T00:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KAKRgTH8k8Okyc4Lz6Ch6En1jp/FHQYRcR3QsmkPqK0/CZL//9ujExddGBv/KuxYVfQUekMCDy04ym3WHmBnDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:10:25.470637Z"},"content_sha256":"6eeb99d7f8dbc15c2b4c9f9d8899f4ea9e0978d087c959ef73d1c820a16c06cd","schema_version":"1.0","event_id":"sha256:6eeb99d7f8dbc15c2b4c9f9d8899f4ea9e0978d087c959ef73d1c820a16c06cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N2Y3MGNKVUHBHTQEPHBFNWADXG/bundle.json","state_url":"https://pith.science/pith/N2Y3MGNKVUHBHTQEPHBFNWADXG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N2Y3MGNKVUHBHTQEPHBFNWADXG/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":"2026-06-09T01:10:25Z","links":{"resolver":"https://pith.science/pith/N2Y3MGNKVUHBHTQEPHBFNWADXG","bundle":"https://pith.science/pith/N2Y3MGNKVUHBHTQEPHBFNWADXG/bundle.json","state":"https://pith.science/pith/N2Y3MGNKVUHBHTQEPHBFNWADXG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N2Y3MGNKVUHBHTQEPHBFNWADXG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:N2Y3MGNKVUHBHTQEPHBFNWADXG","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":"9bf406264e7f612dd42f0fcdb5c5677a3e5d93f09386329e94c59ac5c908de95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T14:33:40Z","title_canon_sha256":"a066d1549e345d0973f312e799acb3d1c3481bd3d8fecf175ba630c083a6f893"},"schema_version":"1.0","source":{"id":"1610.06432","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06432","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06432v2","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06432","created_at":"2026-05-18T00:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"N2Y3MGNKVUHB","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"N2Y3MGNKVUHBHTQE","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"N2Y3MGNK","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:6eeb99d7f8dbc15c2b4c9f9d8899f4ea9e0978d087c959ef73d1c820a16c06cd","target":"graph","created_at":"2026-05-18T00:45:58Z","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 study the existence of $(v,m,k,\\lambda)$-strong external difference families (SEDFs). We use character-theoretic techniques to show that no SEDF exists when $v$ is prime, $k > 1$ and $m > 2$. In the case where $v$ is the product of two distinct odd primes, some necessary conditions are derived, which can be used to rule out certain parameter sets. Further, we show that, when $m=3$ or $4$ and $v > m$, a $(v,m,k,\\lambda)$-SEDF does not exist.","authors_text":"Douglas R. Stinson, William J. Martin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T14:33:40Z","title":"Some Nonexistence Results for Strong External Difference Families Using Character Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06432","kind":"arxiv","version":2},"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":"sha256:65b4d5d8c0358c4c23e3510fa0bfe107b8ac1c643bd3366316e6b90df5513d3c","target":"record","created_at":"2026-05-18T00:45:58Z","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":"9bf406264e7f612dd42f0fcdb5c5677a3e5d93f09386329e94c59ac5c908de95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-10-20T14:33:40Z","title_canon_sha256":"a066d1549e345d0973f312e799acb3d1c3481bd3d8fecf175ba630c083a6f893"},"schema_version":"1.0","source":{"id":"1610.06432","kind":"arxiv","version":2}},"canonical_sha256":"6eb1b619aaad0e13ce0479c256d803b99461f67e02cd34881bf36f5020f99d15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6eb1b619aaad0e13ce0479c256d803b99461f67e02cd34881bf36f5020f99d15","first_computed_at":"2026-05-18T00:45:58.531702Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:58.531702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r60snjyWl9WutLyb4v/YDWSpIya/psKR49TVsuzATOx+ZZauUkaS5YeHPMurvm8K1kQsDQgvRWBcFIqlvCXxAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:58.532170Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.06432","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65b4d5d8c0358c4c23e3510fa0bfe107b8ac1c643bd3366316e6b90df5513d3c","sha256:6eeb99d7f8dbc15c2b4c9f9d8899f4ea9e0978d087c959ef73d1c820a16c06cd"],"state_sha256":"ff1bb29c6e7bc5eead024ae8992dadd5e2a05559cbd86ea7117d51f6c20dad28"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YzjM3JmmWReFtAkSUT0Sa4TQnM9CsqBhNW/TyhCPh/UlUptXjJWDwhBZQL/3aRyZMh8qjzqggXOgrdlFnRNdCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T01:10:25.472507Z","bundle_sha256":"bd3713b5c77ab59b526b4fc79e65bc1e0f0f424f69ad922ee4fce49e0ce9a479"}}