{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:CYIN42Y7ZTES7Z3722YVISWJA7","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":"63aad7cc5e78c22811bce65f5e9359345cbc864816a636538654f1ae6797d140","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-08T20:02:34Z","title_canon_sha256":"19ff82281e6bd746775710921efc15bb5b20081039c480ea481879c48f8439d9"},"schema_version":"1.0","source":{"id":"1904.04333","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.04333","created_at":"2026-05-17T23:48:59Z"},{"alias_kind":"arxiv_version","alias_value":"1904.04333v1","created_at":"2026-05-17T23:48:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.04333","created_at":"2026-05-17T23:48:59Z"},{"alias_kind":"pith_short_12","alias_value":"CYIN42Y7ZTES","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"CYIN42Y7ZTES7Z37","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"CYIN42Y7","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:7a92064737b393744d495801f1876a30cc88561578c9e93fca74d498c7407345","target":"graph","created_at":"2026-05-17T23:48:59Z","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":"In this paper we consider self-dual NRT-codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman (NRT-metric). We use polynomial invariant theory to describe the shape enumerator of a binary self-dual, doubly even self-dual, and doubly-doubly even self dual NRT-code $C\\subseteq M_{n,2}(\\mathbb{F}_{2})$. Motivated by these results we describe the number of invariant polinomials that we must find to describe the shape enumerator of a self-dual NRT-code of $M_{n,s}(\\mathbb{F}_{2})$. We define the ordered flip of a matrix $A\\in M_{k,ns}(\\mathbb{F}_{q})$","authors_text":"Marcelo Muniz Silva Alves, Welington Santos","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-08T20:02:34Z","title":"Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.04333","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":"sha256:451c1c379bffaf385de8623e349e9fdaff9a068965dc8b755de3722967f0f0f1","target":"record","created_at":"2026-05-17T23:48:59Z","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":"63aad7cc5e78c22811bce65f5e9359345cbc864816a636538654f1ae6797d140","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2019-04-08T20:02:34Z","title_canon_sha256":"19ff82281e6bd746775710921efc15bb5b20081039c480ea481879c48f8439d9"},"schema_version":"1.0","source":{"id":"1904.04333","kind":"arxiv","version":1}},"canonical_sha256":"1610de6b1fccc92fe77fd6b1544ac907e2cdb30a0916c6aa708898bae33a7772","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1610de6b1fccc92fe77fd6b1544ac907e2cdb30a0916c6aa708898bae33a7772","first_computed_at":"2026-05-17T23:48:59.258573Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:59.258573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rfxEOByVM5Oir6xUGdiXqVMMW06OvoUcdcrJ/r4sCDroy2a/Bkxxmr+QQVn73/Y9tEYzZbsw8nCSIYoiBpgJAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:59.259137Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.04333","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:451c1c379bffaf385de8623e349e9fdaff9a068965dc8b755de3722967f0f0f1","sha256:7a92064737b393744d495801f1876a30cc88561578c9e93fca74d498c7407345"],"state_sha256":"ddd595c17a9e345719526de108ded7c5e0adc48d0c44f27691886c9c25015913"}