{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:5QX5JSULUI3ORX35XZUQPKHCKI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"961d937da209de2dc11e4f7cd587cecdb2d11bc53d6e9dd5664354927ec02cbf","cross_cats_sorted":["math.AC","math.CO","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-06-22T12:22:32Z","title_canon_sha256":"e1d52b7899d98b14ed322a97879989811feb41c6e143db2601061f5a7b52bdd9"},"schema_version":"1.0","source":{"id":"1206.5124","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5124","created_at":"2026-05-18T01:29:38Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5124v2","created_at":"2026-05-18T01:29:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5124","created_at":"2026-05-18T01:29:38Z"},{"alias_kind":"pith_short_12","alias_value":"5QX5JSULUI3O","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"5QX5JSULUI3ORX35","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"5QX5JSUL","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:a60b10c1cfbf5d41e9df9122a00bce347d4d2c210c6ffe983f64bc0c5faa9110","target":"graph","created_at":"2026-05-18T01:29:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal $I_+(\\mathcal C)$ to an arbitrary linear code. The binomials involved in the reduced Gr\\\"obner basis of such an ideal relative to a degree-compatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of $I_+(\\mathcal C)$ provide a universal test-set which turns out to be a set containing th","authors_text":"Edgar Mart\\'inez-Moro, Emilio Su\\'arez-Canedo, Irene M\\'arquez-Corbella","cross_cats":["math.AC","math.CO","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-06-22T12:22:32Z","title":"On the ideal associated to a linear code"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5124","kind":"arxiv","version":2},"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":"sha256:53e7782584fefc657c49c02c08b3a3005ec7a7d1bea4784c4cfa1a047078aeb3","target":"record","created_at":"2026-05-18T01:29:38Z","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":"961d937da209de2dc11e4f7cd587cecdb2d11bc53d6e9dd5664354927ec02cbf","cross_cats_sorted":["math.AC","math.CO","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-06-22T12:22:32Z","title_canon_sha256":"e1d52b7899d98b14ed322a97879989811feb41c6e143db2601061f5a7b52bdd9"},"schema_version":"1.0","source":{"id":"1206.5124","kind":"arxiv","version":2}},"canonical_sha256":"ec2fd4ca8ba236e8df7dbe6907a8e252237421e0cfa0fe4ba52cafc1fec3e9ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec2fd4ca8ba236e8df7dbe6907a8e252237421e0cfa0fe4ba52cafc1fec3e9ad","first_computed_at":"2026-05-18T01:29:38.665259Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:38.665259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FVtQGYKsrhNqha3QDvFjn9R2GYpP/TKifh7bpt3hRU13wyQ1OfbhHvHU0z5G9mmoOivR1iMpXfTizZcoxOCuBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:38.665770Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5124","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53e7782584fefc657c49c02c08b3a3005ec7a7d1bea4784c4cfa1a047078aeb3","sha256:a60b10c1cfbf5d41e9df9122a00bce347d4d2c210c6ffe983f64bc0c5faa9110"],"state_sha256":"bdf4d67043ea8a9288110cfc34f145d7ad6b4a126b379c9307a350d99e77a2fb"}