{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:GGNXN6WPAQEP4IQWANGYSTOEIR","short_pith_number":"pith:GGNXN6WP","canonical_record":{"source":{"id":"1608.03027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-10T02:10:44Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"d564ebd2e17fee451f2b581fe0f1880707e0a81622205d0dd56f80eb9c642f85","abstract_canon_sha256":"068c811526843991eb3b0c5afebb841ce587277c6828e5b19f306fa9ffc53581"},"schema_version":"1.0"},"canonical_sha256":"319b76facf0408fe2216034d894dc4445b9cc009bec7b3144e0ad2cf07305e92","source":{"kind":"arxiv","id":"1608.03027","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.03027","created_at":"2026-05-18T01:09:30Z"},{"alias_kind":"arxiv_version","alias_value":"1608.03027v1","created_at":"2026-05-18T01:09:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.03027","created_at":"2026-05-18T01:09:30Z"},{"alias_kind":"pith_short_12","alias_value":"GGNXN6WPAQEP","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GGNXN6WPAQEP4IQW","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GGNXN6WP","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:GGNXN6WPAQEP4IQWANGYSTOEIR","target":"record","payload":{"canonical_record":{"source":{"id":"1608.03027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-10T02:10:44Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"d564ebd2e17fee451f2b581fe0f1880707e0a81622205d0dd56f80eb9c642f85","abstract_canon_sha256":"068c811526843991eb3b0c5afebb841ce587277c6828e5b19f306fa9ffc53581"},"schema_version":"1.0"},"canonical_sha256":"319b76facf0408fe2216034d894dc4445b9cc009bec7b3144e0ad2cf07305e92","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:30.453265Z","signature_b64":"woCQHRFmHZNC9YVJh2hkqtXw2+ZUw5gV61QW/QFQ73/U7uKbbrPoXaGVjatyspGMyOSZTL6jTh8G2X0HC/JZCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"319b76facf0408fe2216034d894dc4445b9cc009bec7b3144e0ad2cf07305e92","last_reissued_at":"2026-05-18T01:09:30.452833Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:30.452833Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.03027","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":"2026-05-18T01:09:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x54WrU//6Ktln4gK+nT2+XcJpjaCq/b5tm1ItgB8emvvStqgcLbgsv2I0QoXbHXjM657MvlOWl+jtOGDZMdcBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T05:29:52.107413Z"},"content_sha256":"f66b61e865c467dfaf93d34864826728595d064d3d86b0aeaf37e8caf40011d9","schema_version":"1.0","event_id":"sha256:f66b61e865c467dfaf93d34864826728595d064d3d86b0aeaf37e8caf40011d9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:GGNXN6WPAQEP4IQWANGYSTOEIR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dimensions of three types of BCH codes over GF(q)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Chengju Li, Cunsheng Ding, Hao Liu","submitted_at":"2016-08-10T02:10:44Z","abstract_excerpt":"BCH codes have been studied for over fifty years and widely employed in consumer devices, communication systems, and data storage systems. However, the dimension of BCH codes is settled only for a very small number of cases. In this paper, we study the dimensions of BCH codes over finite fields with three types of lengths $n$, namely $n=q^m-1$, $n=(q^m-1)/(q-1)$ and $n=q^m+1$. For narrow-sense primitive BCH codes with designed distance $\\delta$, we investigate their dimensions for $\\delta$ in the range $1\\le \\delta \\le q^{\\lceil\\frac{m}{2}\\rceil+1}$. For non-narrow sense primitive BCH codes, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03027","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":"2026-05-18T01:09:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t3TLjlhXzIsSn2c7VA90TDUE6K8rBQTHcNPnaJkNRqxw6s0/N079qe1ztTASZFoD4VXAwfBtYhjYVM8cv9n0Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T05:29:52.108099Z"},"content_sha256":"4bfa5dd02c930ee35d77df293971f17782b18af7a0dc9873505e9e401069076f","schema_version":"1.0","event_id":"sha256:4bfa5dd02c930ee35d77df293971f17782b18af7a0dc9873505e9e401069076f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GGNXN6WPAQEP4IQWANGYSTOEIR/bundle.json","state_url":"https://pith.science/pith/GGNXN6WPAQEP4IQWANGYSTOEIR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GGNXN6WPAQEP4IQWANGYSTOEIR/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":"2026-06-06T05:29:52Z","links":{"resolver":"https://pith.science/pith/GGNXN6WPAQEP4IQWANGYSTOEIR","bundle":"https://pith.science/pith/GGNXN6WPAQEP4IQWANGYSTOEIR/bundle.json","state":"https://pith.science/pith/GGNXN6WPAQEP4IQWANGYSTOEIR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GGNXN6WPAQEP4IQWANGYSTOEIR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:GGNXN6WPAQEP4IQWANGYSTOEIR","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":"068c811526843991eb3b0c5afebb841ce587277c6828e5b19f306fa9ffc53581","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-10T02:10:44Z","title_canon_sha256":"d564ebd2e17fee451f2b581fe0f1880707e0a81622205d0dd56f80eb9c642f85"},"schema_version":"1.0","source":{"id":"1608.03027","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.03027","created_at":"2026-05-18T01:09:30Z"},{"alias_kind":"arxiv_version","alias_value":"1608.03027v1","created_at":"2026-05-18T01:09:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.03027","created_at":"2026-05-18T01:09:30Z"},{"alias_kind":"pith_short_12","alias_value":"GGNXN6WPAQEP","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GGNXN6WPAQEP4IQW","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GGNXN6WP","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:4bfa5dd02c930ee35d77df293971f17782b18af7a0dc9873505e9e401069076f","target":"graph","created_at":"2026-05-18T01:09:30Z","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":"BCH codes have been studied for over fifty years and widely employed in consumer devices, communication systems, and data storage systems. However, the dimension of BCH codes is settled only for a very small number of cases. In this paper, we study the dimensions of BCH codes over finite fields with three types of lengths $n$, namely $n=q^m-1$, $n=(q^m-1)/(q-1)$ and $n=q^m+1$. For narrow-sense primitive BCH codes with designed distance $\\delta$, we investigate their dimensions for $\\delta$ in the range $1\\le \\delta \\le q^{\\lceil\\frac{m}{2}\\rceil+1}$. For non-narrow sense primitive BCH codes, w","authors_text":"Chengju Li, Cunsheng Ding, Hao Liu","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-10T02:10:44Z","title":"Dimensions of three types of BCH codes over GF(q)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03027","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:f66b61e865c467dfaf93d34864826728595d064d3d86b0aeaf37e8caf40011d9","target":"record","created_at":"2026-05-18T01:09:30Z","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":"068c811526843991eb3b0c5afebb841ce587277c6828e5b19f306fa9ffc53581","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-08-10T02:10:44Z","title_canon_sha256":"d564ebd2e17fee451f2b581fe0f1880707e0a81622205d0dd56f80eb9c642f85"},"schema_version":"1.0","source":{"id":"1608.03027","kind":"arxiv","version":1}},"canonical_sha256":"319b76facf0408fe2216034d894dc4445b9cc009bec7b3144e0ad2cf07305e92","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"319b76facf0408fe2216034d894dc4445b9cc009bec7b3144e0ad2cf07305e92","first_computed_at":"2026-05-18T01:09:30.452833Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:30.452833Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"woCQHRFmHZNC9YVJh2hkqtXw2+ZUw5gV61QW/QFQ73/U7uKbbrPoXaGVjatyspGMyOSZTL6jTh8G2X0HC/JZCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:30.453265Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.03027","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f66b61e865c467dfaf93d34864826728595d064d3d86b0aeaf37e8caf40011d9","sha256:4bfa5dd02c930ee35d77df293971f17782b18af7a0dc9873505e9e401069076f"],"state_sha256":"67c484ed326fd8585ba332d3f26ace2c213cc2975ba8cdde4d9c90351a23d667"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y2g9Uv5m5Nzer3krmlTsy+FgYGwJWdU2ERZPRZYx8tc5f2pHHiR9n4fpldx22hAIC/WbJdgouNITd4cA2sQcDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T05:29:52.111737Z","bundle_sha256":"853cb679e870e75eee0f9f9baea2bcfd4e8588a5b62ff941b62948ac5cd971ba"}}