{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:HNZBGFWA4BW2RZGS2R7FD4AYO3","short_pith_number":"pith:HNZBGFWA","canonical_record":{"source":{"id":"1303.6985","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-27T21:29:16Z","cross_cats_sorted":[],"title_canon_sha256":"55f6f08f742f2e72570144e0a2b3aa7ba6a3382bd61be2d5c9e5eba19a3b653c","abstract_canon_sha256":"508c3ba8f4a29f53555c9bdfcf64db2e14050ed9a8c873e40c546a8b40ff6e63"},"schema_version":"1.0"},"canonical_sha256":"3b721316c0e06da8e4d2d47e51f01876e29dcabd275e6abd02a2e5947efab4c7","source":{"kind":"arxiv","id":"1303.6985","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6985","created_at":"2026-05-18T03:29:38Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6985v1","created_at":"2026-05-18T03:29:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6985","created_at":"2026-05-18T03:29:38Z"},{"alias_kind":"pith_short_12","alias_value":"HNZBGFWA4BW2","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HNZBGFWA4BW2RZGS","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HNZBGFWA","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:HNZBGFWA4BW2RZGS2R7FD4AYO3","target":"record","payload":{"canonical_record":{"source":{"id":"1303.6985","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-27T21:29:16Z","cross_cats_sorted":[],"title_canon_sha256":"55f6f08f742f2e72570144e0a2b3aa7ba6a3382bd61be2d5c9e5eba19a3b653c","abstract_canon_sha256":"508c3ba8f4a29f53555c9bdfcf64db2e14050ed9a8c873e40c546a8b40ff6e63"},"schema_version":"1.0"},"canonical_sha256":"3b721316c0e06da8e4d2d47e51f01876e29dcabd275e6abd02a2e5947efab4c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:38.958168Z","signature_b64":"lAXN9fENsy2DMr3yMBthG+zlh71suRlX+SDG9NGt1IwfRhYdKR3Rk3gIzmxWV3qUsJhHh6wAoljIF8nDBT33DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b721316c0e06da8e4d2d47e51f01876e29dcabd275e6abd02a2e5947efab4c7","last_reissued_at":"2026-05-18T03:29:38.957301Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:38.957301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.6985","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":"2026-05-18T03:29:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bGZ6APWR/ogrsWKEJULKwaJBE0rZATUb07VhMtAuoowR/Hfe+yPB9v4JYjI226E2cwwOsJeiucfDUnnAcrF/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T23:10:10.023583Z"},"content_sha256":"6b5ef2cf194034f720e50b8b8f85be4bddbce9b7b3a5e400f6ad823f4261411e","schema_version":"1.0","event_id":"sha256:6b5ef2cf194034f720e50b8b8f85be4bddbce9b7b3a5e400f6ad823f4261411e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:HNZBGFWA4BW2RZGS2R7FD4AYO3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting The Generator Matrices of $\\mathbb{Z}_{2}\\mathbb{Z}_{8}$-Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Irfan Siap, Ismail Aydogdu","submitted_at":"2013-03-27T21:29:16Z","abstract_excerpt":"In this paper, we count the number of matrices whose rows generate different $\\mathbb{Z}_2\\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). The MGN formula by specialization leads to the well known formula for the number of binary codes and the number of codes over $\\mathbb{Z}_8,$ and for additive $\\mathbb{Z}_2\\mathbb{Z}_4$ codes. Also, we conclud"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6985","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":"2026-05-18T03:29:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gHxCqHR0I9mLmLAxHJSfjRvDdIO5/xVRMupc0cojQsON6YR9xDyoAxqLzuOCbeV1nS/MSQTNQb3Fy5Hfe8QuBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T23:10:10.023921Z"},"content_sha256":"c7540bc1ffca2bdb482667c1f57b54a73a02d085a74833df50fe432e4252d501","schema_version":"1.0","event_id":"sha256:c7540bc1ffca2bdb482667c1f57b54a73a02d085a74833df50fe432e4252d501"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HNZBGFWA4BW2RZGS2R7FD4AYO3/bundle.json","state_url":"https://pith.science/pith/HNZBGFWA4BW2RZGS2R7FD4AYO3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HNZBGFWA4BW2RZGS2R7FD4AYO3/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":"2026-06-26T23:10:10Z","links":{"resolver":"https://pith.science/pith/HNZBGFWA4BW2RZGS2R7FD4AYO3","bundle":"https://pith.science/pith/HNZBGFWA4BW2RZGS2R7FD4AYO3/bundle.json","state":"https://pith.science/pith/HNZBGFWA4BW2RZGS2R7FD4AYO3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HNZBGFWA4BW2RZGS2R7FD4AYO3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HNZBGFWA4BW2RZGS2R7FD4AYO3","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":"508c3ba8f4a29f53555c9bdfcf64db2e14050ed9a8c873e40c546a8b40ff6e63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-27T21:29:16Z","title_canon_sha256":"55f6f08f742f2e72570144e0a2b3aa7ba6a3382bd61be2d5c9e5eba19a3b653c"},"schema_version":"1.0","source":{"id":"1303.6985","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6985","created_at":"2026-05-18T03:29:38Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6985v1","created_at":"2026-05-18T03:29:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6985","created_at":"2026-05-18T03:29:38Z"},{"alias_kind":"pith_short_12","alias_value":"HNZBGFWA4BW2","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HNZBGFWA4BW2RZGS","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HNZBGFWA","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:c7540bc1ffca2bdb482667c1f57b54a73a02d085a74833df50fe432e4252d501","target":"graph","created_at":"2026-05-18T03:29:38Z","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 count the number of matrices whose rows generate different $\\mathbb{Z}_2\\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). The MGN formula by specialization leads to the well known formula for the number of binary codes and the number of codes over $\\mathbb{Z}_8,$ and for additive $\\mathbb{Z}_2\\mathbb{Z}_4$ codes. Also, we conclud","authors_text":"Irfan Siap, Ismail Aydogdu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-27T21:29:16Z","title":"Counting The Generator Matrices of $\\mathbb{Z}_{2}\\mathbb{Z}_{8}$-Codes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6985","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":"sha256:6b5ef2cf194034f720e50b8b8f85be4bddbce9b7b3a5e400f6ad823f4261411e","target":"record","created_at":"2026-05-18T03:29:38Z","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":"508c3ba8f4a29f53555c9bdfcf64db2e14050ed9a8c873e40c546a8b40ff6e63","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-27T21:29:16Z","title_canon_sha256":"55f6f08f742f2e72570144e0a2b3aa7ba6a3382bd61be2d5c9e5eba19a3b653c"},"schema_version":"1.0","source":{"id":"1303.6985","kind":"arxiv","version":1}},"canonical_sha256":"3b721316c0e06da8e4d2d47e51f01876e29dcabd275e6abd02a2e5947efab4c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3b721316c0e06da8e4d2d47e51f01876e29dcabd275e6abd02a2e5947efab4c7","first_computed_at":"2026-05-18T03:29:38.957301Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:38.957301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lAXN9fENsy2DMr3yMBthG+zlh71suRlX+SDG9NGt1IwfRhYdKR3Rk3gIzmxWV3qUsJhHh6wAoljIF8nDBT33DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:38.958168Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.6985","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6b5ef2cf194034f720e50b8b8f85be4bddbce9b7b3a5e400f6ad823f4261411e","sha256:c7540bc1ffca2bdb482667c1f57b54a73a02d085a74833df50fe432e4252d501"],"state_sha256":"7b75fde226d39c492a3f5c9d22271d10338f3ebc74e8c85705248e1635816046"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"59lInEqoJNiIHQhq7fvCyoisSErdr+vV6FAUe2ER52YM3ecmZ5cpsko/MwWuyWMu5tG3JO2erpBeX5GJc5LVAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T23:10:10.026121Z","bundle_sha256":"6cd281ed3243a960f4c11302dce58229501e1945ae297fe845ef386a36eba540"}}