{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:6GZZWUQQ4Y5LU4NXLSQLQ25Z2M","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":"1e28de7a496cc033c550145049464e70205b839fe9a25a76685fdde6d997940f","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-12-01T03:26:09Z","title_canon_sha256":"195284e0f2a8252b605c986c28225267973d6fc2bb763836b3d50a2f091fcc4b"},"schema_version":"1.0","source":{"id":"1612.00126","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.00126","created_at":"2026-05-18T00:53:23Z"},{"alias_kind":"arxiv_version","alias_value":"1612.00126v1","created_at":"2026-05-18T00:53:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.00126","created_at":"2026-05-18T00:53:23Z"},{"alias_kind":"pith_short_12","alias_value":"6GZZWUQQ4Y5L","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"6GZZWUQQ4Y5LU4NX","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"6GZZWUQQ","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:7b44476d2ffed356fa7b229734758c9ef6a31799d2dddbc6627288e1c762f086","target":"graph","created_at":"2026-05-18T00:53:23Z","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":"We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring $R=\\mathbb{F}_2 +v\\mathbb{F}_2 +v^2\\mathbb{F}_2 +v^3\\mathbb{F}_2 +v^4\\mathbb{F}_2,$ where $v^5=1.$ The same ring occurs in the quintic construction of binary quasi-cyclic codes. %The length of these codes depends on the degree $m$ of ring extension. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Given a linear Gray map, we obtain three families of binary abelian codes with few weights. In particular, we obtain a class ","authors_text":"Minjia Shi, Patrick Sol\\'e, Yan Liu","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-12-01T03:26:09Z","title":"Three-weight codes and the quintic construction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00126","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":"sha256:3c5eb0b67b5180aa86ddb00bdf4ff15c2debbfa047d2a57f189378f8203ad8d6","target":"record","created_at":"2026-05-18T00:53:23Z","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":"1e28de7a496cc033c550145049464e70205b839fe9a25a76685fdde6d997940f","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-12-01T03:26:09Z","title_canon_sha256":"195284e0f2a8252b605c986c28225267973d6fc2bb763836b3d50a2f091fcc4b"},"schema_version":"1.0","source":{"id":"1612.00126","kind":"arxiv","version":1}},"canonical_sha256":"f1b39b5210e63aba71b75ca0b86bb9d305271ee65ac517384a87a659220ca25b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f1b39b5210e63aba71b75ca0b86bb9d305271ee65ac517384a87a659220ca25b","first_computed_at":"2026-05-18T00:53:23.995237Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:23.995237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bnx3VswcBeES7z+IHrN7okyDVb65w1Bp6QwWdirBH+GK4A5A6qRfIdUqCnKIQPSNxBWOZTuPiJIxe96lsKMnAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:23.995674Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.00126","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c5eb0b67b5180aa86ddb00bdf4ff15c2debbfa047d2a57f189378f8203ad8d6","sha256:7b44476d2ffed356fa7b229734758c9ef6a31799d2dddbc6627288e1c762f086"],"state_sha256":"4487cf3180a4708d70cf095bfc33fc9bfbcbda8a474eca4642fd33d29b699950"}