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Let each $g_i$ be of order $n_i$ in $\\Bbb F_r^*$ and $\\gcd(n_i, n_j)=1$ for $1\\leq i \\neq j \\leq u$.\n  We define a cyclic code over $\\Bbb F_q$ by\n  $$\\mathcal C_{(q, m, n_1,n_2, ..., n_u)}=\\{c(a_1, a_2, ..., a_u) : a_1, a_2, ..., a_u \\in \\Bbb F_r\\},$$ where\n  $$c(a_1, a_2, ..., a_u)=({Tr}_{r/q}(\\sum_{i=1}^ua_ig_i^0), ..., {Tr}_{r/q}(\\sum_{i=1}^ua_ig_i^{n-1}))$$ and $n=n_1n_2... n_u$. In this paper, we present a method to compute the weights of $\\mathcal C_{(q, m, n_1,n_2, ..., n_u)}$. Further, we determine the weight dis"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1306.5809","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2013-06-24T23:35:12Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"28c04e156234b5dc084a69efff8927ab2987a2c4c55af067f45a9c80a2d7e4de","abstract_canon_sha256":"aac382b2c644a1cad746449d79054bd373d00d8d16f24e6d02ebd65e962249d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:34.465764Z","signature_b64":"bATPQKHKJ3BGfij8XVMZw9s3U3FjYVQlXF05Gqtp9DHm+hsOTIhKAyCoMpNx4/fsbH+c/vvea+SaL88++ynJCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b075265664a59a4ab753eb74da8857a5ed5c8aae1f81a5b5830434e659205f15","last_reissued_at":"2026-05-18T03:19:34.465410Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:34.465410Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weight distributions of cyclic codes with respect to pairwise coprime order elements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Chengju Li, Fengwei Li, Qin Yue","submitted_at":"2013-06-24T23:35:12Z","abstract_excerpt":"Let $\\Bbb F_r$ be an extension of a finite field $\\Bbb F_q$ with $r=q^m$. Let each $g_i$ be of order $n_i$ in $\\Bbb F_r^*$ and $\\gcd(n_i, n_j)=1$ for $1\\leq i \\neq j \\leq u$.\n  We define a cyclic code over $\\Bbb F_q$ by\n  $$\\mathcal C_{(q, m, n_1,n_2, ..., n_u)}=\\{c(a_1, a_2, ..., a_u) : a_1, a_2, ..., a_u \\in \\Bbb F_r\\},$$ where\n  $$c(a_1, a_2, ..., a_u)=({Tr}_{r/q}(\\sum_{i=1}^ua_ig_i^0), ..., {Tr}_{r/q}(\\sum_{i=1}^ua_ig_i^{n-1}))$$ and $n=n_1n_2... n_u$. In this paper, we present a method to compute the weights of $\\mathcal C_{(q, m, n_1,n_2, ..., n_u)}$. Further, we determine the weight dis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5809","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1306.5809","created_at":"2026-05-18T03:19:34.465462+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.5809v2","created_at":"2026-05-18T03:19:34.465462+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.5809","created_at":"2026-05-18T03:19:34.465462+00:00"},{"alias_kind":"pith_short_12","alias_value":"WB2SMVTEUWNE","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"WB2SMVTEUWNEVN2T","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"WB2SMVTE","created_at":"2026-05-18T12:28:04.890932+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX","json":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX.json","graph_json":"https://pith.science/api/pith-number/WB2SMVTEUWNEVN2T5N2NVCCXUX/graph.json","events_json":"https://pith.science/api/pith-number/WB2SMVTEUWNEVN2T5N2NVCCXUX/events.json","paper":"https://pith.science/paper/WB2SMVTE"},"agent_actions":{"view_html":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX","download_json":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX.json","view_paper":"https://pith.science/paper/WB2SMVTE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.5809&json=true","fetch_graph":"https://pith.science/api/pith-number/WB2SMVTEUWNEVN2T5N2NVCCXUX/graph.json","fetch_events":"https://pith.science/api/pith-number/WB2SMVTEUWNEVN2T5N2NVCCXUX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX/action/storage_attestation","attest_author":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX/action/author_attestation","sign_citation":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX/action/citation_signature","submit_replication":"https://pith.science/pith/WB2SMVTEUWNEVN2T5N2NVCCXUX/action/replication_record"}},"created_at":"2026-05-18T03:19:34.465462+00:00","updated_at":"2026-05-18T03:19:34.465462+00:00"}