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Shparlinski, Zhixiong Chen","submitted_at":"2013-10-01T02:14:04Z","abstract_excerpt":"For a set\n  $\\cM=\\{-\\mu,-\\mu+1,\\ldots, \\lambda\\}\\setminus\\{0\\}$ with non-negative integers $\\lambda,\\mu<q$ not both 0, a subset $\\cS$ of the residue class ring $\\Z_q$ modulo an integer $q\\ge 1$ is called a $(\\lambda,\\mu;q)$-\\emph{covering set} if $$ \\cM \\cS=\\{ms \\bmod q : m\\in \\cM,\\ s\\in \\cS\\}=\\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(\\lambda,\\mu;q)$-covering set $\\cS$ which is of the size $q^{1 + o(1)}\\max\\{\\lambda,\\mu\\}^{-1/2}$ for almost all integers $q\\ge 1$ and of optimal size $p\\max\\{\\lambda,\\"},"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":"1310.0120","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2013-10-01T02:14:04Z","cross_cats_sorted":["math.IT","math.NT"],"title_canon_sha256":"01aabbbd109e20fdcb3f4a229cdcd678ce52f6b168201c4387cd1d04877a1e09","abstract_canon_sha256":"bd4e6e14c7c82064858dbd9b0945a64eedef44b634890f528cd4f1458e0aeb35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:45.556256Z","signature_b64":"x0SXJ6qIQk5rY9lceHhoMLkTi0xn1hwAanj5FAtxLTx5l0e3yz9Y5EiQk8Xl97OzmC2iP/TWKDrdFQu4kYxMCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c529a07cc8b8b84b9ce99ef7b3ca014f4d3b6ca0f0b85b8283f85ad1f2380db3","last_reissued_at":"2026-05-18T03:11:45.555679Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:45.555679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Covering sets for limited-magnitude errors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.NT"],"primary_cat":"cs.IT","authors_text":"Arne Winterhof, Igor E. Shparlinski, Zhixiong Chen","submitted_at":"2013-10-01T02:14:04Z","abstract_excerpt":"For a set\n  $\\cM=\\{-\\mu,-\\mu+1,\\ldots, \\lambda\\}\\setminus\\{0\\}$ with non-negative integers $\\lambda,\\mu<q$ not both 0, a subset $\\cS$ of the residue class ring $\\Z_q$ modulo an integer $q\\ge 1$ is called a $(\\lambda,\\mu;q)$-\\emph{covering set} if $$ \\cM \\cS=\\{ms \\bmod q : m\\in \\cM,\\ s\\in \\cS\\}=\\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(\\lambda,\\mu;q)$-covering set $\\cS$ which is of the size $q^{1 + o(1)}\\max\\{\\lambda,\\mu\\}^{-1/2}$ for almost all integers $q\\ge 1$ and of optimal size $p\\max\\{\\lambda,\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0120","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1310.0120","created_at":"2026-05-18T03:11:45.555779+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.0120v1","created_at":"2026-05-18T03:11:45.555779+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.0120","created_at":"2026-05-18T03:11:45.555779+00:00"},{"alias_kind":"pith_short_12","alias_value":"YUU2A7GIXC4E","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"YUU2A7GIXC4EXHHJ","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"YUU2A7GI","created_at":"2026-05-18T12:28:09.283467+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/YUU2A7GIXC4EXHHJT333HSQBJ5","json":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5.json","graph_json":"https://pith.science/api/pith-number/YUU2A7GIXC4EXHHJT333HSQBJ5/graph.json","events_json":"https://pith.science/api/pith-number/YUU2A7GIXC4EXHHJT333HSQBJ5/events.json","paper":"https://pith.science/paper/YUU2A7GI"},"agent_actions":{"view_html":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5","download_json":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5.json","view_paper":"https://pith.science/paper/YUU2A7GI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.0120&json=true","fetch_graph":"https://pith.science/api/pith-number/YUU2A7GIXC4EXHHJT333HSQBJ5/graph.json","fetch_events":"https://pith.science/api/pith-number/YUU2A7GIXC4EXHHJT333HSQBJ5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5/action/storage_attestation","attest_author":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5/action/author_attestation","sign_citation":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5/action/citation_signature","submit_replication":"https://pith.science/pith/YUU2A7GIXC4EXHHJT333HSQBJ5/action/replication_record"}},"created_at":"2026-05-18T03:11:45.555779+00:00","updated_at":"2026-05-18T03:11:45.555779+00:00"}