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We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the subgroup is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of subgroups of $T_X$ are r"},"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":"1712.00747","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-12-03T10:44:44Z","cross_cats_sorted":["cs.IT","math.AC","math.CO","math.IT"],"title_canon_sha256":"191e49c6441c9214c270f0870fb642f0a36ea71dffddd7cd2a2e5de0b79c397e","abstract_canon_sha256":"7d320d0d351698d56c7d7b53a5bd89fd45c7dcda02d058426ad0858cb7d5fa19"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:23.878878Z","signature_b64":"wpneAKa2ZrDOeiHkRzVmbnT73tJ0oFf/ge6nNscb4cZ7nbkIkao4lUTmAGBsCd4NosMtyjMNv9fQjiHXaBMwBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7bcc1c780a3f05bf7ea194bdba76e3c4473fb791bbb00933ea1b7b9d0fef2234","last_reissued_at":"2026-05-18T00:04:23.878247Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:23.878247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Toric Codes and Lattice Ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.AC","math.CO","math.IT"],"primary_cat":"math.AG","authors_text":"Mesut \\c{S}ahin","submitted_at":"2017-12-03T10:44:44Z","abstract_excerpt":"Let $X$ be a complete simplicial toric variety over a finite field $\\mathbb{F}_q$ with homogeneous coordinate ring $S=\\mathbb{F}_q[x_1,\\dots,x_r]$ and split torus $T_X\\cong (\\mathbb{F}^*_q)^n$. We prove that vanishing ideal of a subset $Y$ of the torus $T_X$ is a lattice ideal if and only if $Y$ is a subgroup. We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the subgroup is the zero locus of a lattice ideal in the torus. 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