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These codes can be identified as submodules of the $\\mathbb{Z}_4[x]$-module $\\mathbb{Z}_2[x]/(x^\\alpha-1)\\times\\mathbb{Z}_4[x]/(x^\\beta-1)$. Any $\\mathbb{Z}_2\\mathbb{Z}_4$-additive cyclic code ${\\cal C}$ is of the form $\\langle (b("},"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":"1606.01745","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-06-06T13:52:22Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"482cc9e782db45f904b0dd56b7e8c22b2fba56c8cc84b4031cf3bfa88ce62416","abstract_canon_sha256":"6f7de46373b78fec24fca0c0497ce586fdb0a7df669ce04a7814f8847b557565"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:53.672566Z","signature_b64":"Z7WVlc0ebSc4g4cWnUb9r9URnYPQXkTlFpYPAsgipgi10gMmxb3m4sHWv0fAYjyiKqRaG2cKg4h+0RTFpKldCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44fd7d0e60985185d8e45729434c24538b28fc5ba2b15d9b80831fa5acf03202","last_reissued_at":"2026-05-18T01:12:53.672213Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:53.672213Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computing the generator polynomials of $\\mathbb{Z}_2\\mathbb{Z}_4$-additive cyclic codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Cristina Fern\\'andez-C\\'ordoba, Joaquim Borges Ayats, Roger Ten-Valls","submitted_at":"2016-06-06T13:52:22Z","abstract_excerpt":"A ${\\mathbb{Z}}_2{\\mathbb{Z}}_4$-additive code ${\\cal C}\\subseteq{\\mathbb{Z}}_2^\\alpha\\times{\\mathbb{Z}}_4^\\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\\mathbb{Z}}_2$ and the set of ${\\mathbb{Z}}_4$ coordinates, such that any simultaneous cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the $\\mathbb{Z}_4[x]$-module $\\mathbb{Z}_2[x]/(x^\\alpha-1)\\times\\mathbb{Z}_4[x]/(x^\\beta-1)$. 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