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The immunity of $MOD_q$ is lower bounded by $\\lfloor (n+1)/2 \\rfloor$, which is achievable when $n$ is a multiple of $2q$; the immunity of $\\neg MOD_q$ is exactly $\\lfloor (n+q-1)/q \\rfloor$ for every $q$ and $n$. Our result improves the previous bound $\\lfloor \\frac{n}{2(q-1)} \\rfloor$ by Green.\n  We observe how","authors_text":"Chris Beck, Yuan Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-02T17:50:54Z","title":"Represent MOD function by low degree polynomial with unbounded one-sided error"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0713","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:b0323a89867fb03741da14de6f3cd18953530a9393e7bf19899c1ea8384c0969","target":"record","created_at":"2026-05-18T03:29:14Z","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":"9a98fee7c50ad4fe6c408b45be34e2204c18af7e95719b99f3004d8d0ffb18ab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2013-04-02T17:50:54Z","title_canon_sha256":"a3b069ea855a399c4a003a929166a6e6a8f3ea620326b7217d6888302a456422"},"schema_version":"1.0","source":{"id":"1304.0713","kind":"arxiv","version":1}},"canonical_sha256":"1aaa0302efa41e93f1f7a3dc9e93a538239fb3619cedbab7ed3a26f22cd611ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1aaa0302efa41e93f1f7a3dc9e93a538239fb3619cedbab7ed3a26f22cd611ce","first_computed_at":"2026-05-18T03:29:14.556474Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:14.556474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iPAzDugUZMuqp3XHbCDcJllT1hljmI9gyavKU4GquGcwTAi9gMEtBH3dng3kNoNJ15ehRG1rpeJxgOpgDe6UBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:14.557124Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.0713","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0323a89867fb03741da14de6f3cd18953530a9393e7bf19899c1ea8384c0969","sha256:0bf25edfb8c44293708fb469f0b1387107e886ded0fccb097a4f9b796770b2fc"],"state_sha256":"8b211dc20134c176091cd4232679071aaf2db19fd285efc921b707afcd75305b"}