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In this work we explore their local decodability and (tolerant) local testability. While these aspects have been studied extensively when $A_1 = \\cdots = A_n = \\mathbb{F}_q$ are the same finite field, the setting when $A_i$'s are not the full field does not seem to have been explored before.\n  In this work we focu"},"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":"1709.06036","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-09-18T16:40:54Z","cross_cats_sorted":[],"title_canon_sha256":"7b967c27f4e9e4416f80bbdf6d6bd2221cbb4ea1e8043b7a2c58c0ce2cc66c2a","abstract_canon_sha256":"209f1227c83629cd0ee9ee4e48e0cce102a36b23ccf33a046a5014b41ba81abc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:19.621684Z","signature_b64":"rbl41NyilucGGizm1t+kWa+eLINcoDf0nPwYOacB62UjlqNqZWF+ss+dZMbwtZKWZrwb9o+c1ylBT7GTJnSvBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bc44f822c82add31dd0a91ea149b1de3c28be8fb2af41c0f89f3e11b7e58f89","last_reissued_at":"2026-05-17T23:58:19.621054Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:19.621054Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local decoding and testing of polynomials over grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Madhu Sudan, Mitali Bafna, Srikanth Srinivasan","submitted_at":"2017-09-18T16:40:54Z","abstract_excerpt":"The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that $n$-variate polynomials of total degree at most $d$ over grids, i.e. sets of the form $A_1 \\times A_2 \\times \\cdots \\times A_n$, form error-correcting codes (of distance at least $2^{-d}$ provided $\\min_i\\{|A_i|\\}\\geq 2$). 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