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Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$-designs with $t=3,4$. 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Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$-designs with $t=3,4$. In particular, we prove that the codewords of weight $7$ in $\\mathcal{C}_{(q,q+1,4,1)}$ support $4$-designs when $m \\geqslant 5$ is odd and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.10903","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.10903/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2106.10903","created_at":"2026-07-05T02:54:18.703743+00:00"},{"alias_kind":"arxiv_version","alias_value":"2106.10903v2","created_at":"2026-07-05T02:54:18.703743+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.10903","created_at":"2026-07-05T02:54:18.703743+00:00"},{"alias_kind":"pith_short_12","alias_value":"7Y3CB77W7S37","created_at":"2026-07-05T02:54:18.703743+00:00"},{"alias_kind":"pith_short_16","alias_value":"7Y3CB77W7S37WFLD","created_at":"2026-07-05T02:54:18.703743+00:00"},{"alias_kind":"pith_short_8","alias_value":"7Y3CB77W","created_at":"2026-07-05T02:54:18.703743+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/7Y3CB77W7S37WFLDVQRNNFQSFY","json":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY.json","graph_json":"https://pith.science/api/pith-number/7Y3CB77W7S37WFLDVQRNNFQSFY/graph.json","events_json":"https://pith.science/api/pith-number/7Y3CB77W7S37WFLDVQRNNFQSFY/events.json","paper":"https://pith.science/paper/7Y3CB77W"},"agent_actions":{"view_html":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY","download_json":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY.json","view_paper":"https://pith.science/paper/7Y3CB77W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2106.10903&json=true","fetch_graph":"https://pith.science/api/pith-number/7Y3CB77W7S37WFLDVQRNNFQSFY/graph.json","fetch_events":"https://pith.science/api/pith-number/7Y3CB77W7S37WFLDVQRNNFQSFY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY/action/storage_attestation","attest_author":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY/action/author_attestation","sign_citation":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY/action/citation_signature","submit_replication":"https://pith.science/pith/7Y3CB77W7S37WFLDVQRNNFQSFY/action/replication_record"}},"created_at":"2026-07-05T02:54:18.703743+00:00","updated_at":"2026-07-05T02:54:18.703743+00:00"}