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We prove that if the binary Gray image of ${\\cal C}$, $C=\\Phi({\\cal C})$, is a 1-perfect nonlinear code, then ${\\cal C}$ cannot be a ${\\mathbb{Z}}_2{\\mathbb{Z}}_4$-cyclic code except for one case of length $n=15$. Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a ${\\mathbb{Z}}_2{\\mathbb{Z}}_4$-additive 1-perfect code gives an extended 1-perfect code. We also prove that any such code cannot be ${\\mathbb{Z}}_2{\\mathbb{Z}}_4$-cyclic."},"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":"1510.06166","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-21T08:37:34Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"4582705143b7a8077a1b9c0615dc872d4604fa9e1342bdc993cd4c6f095f9655","abstract_canon_sha256":"ed8897a25b113bdbbeda1296529dfbc262fe55886737f33b619180e552d22616"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:37.624267Z","signature_b64":"mU3R7UUL4WNgpZ9gu3lEFQngOoLndRAEuaV4lDavdvS3wA6Nz8GdDryTO86s9u1m8mKUYjHlVozkxCaF76+4Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b9eeba4070122b32631a0869dbae541c5c488ffedbf1cc3556b2e23a30267e5","last_reissued_at":"2026-05-18T01:29:37.623647Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:37.623647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"There is exactly one Z2Z4-cyclic 1-perfect code","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Cristina Fern\\'andez-C\\'ordoba, Joaquim Borges","submitted_at":"2015-10-21T08:37:34Z","abstract_excerpt":"Let ${\\cal C}$ be a ${\\mathbb{Z}}_2{\\mathbb{Z}}_4$-additive code of length $n > 3$. We prove that if the binary Gray image of ${\\cal C}$, $C=\\Phi({\\cal C})$, is a 1-perfect nonlinear code, then ${\\cal C}$ cannot be a ${\\mathbb{Z}}_2{\\mathbb{Z}}_4$-cyclic code except for one case of length $n=15$. Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a ${\\mathbb{Z}}_2{\\mathbb{Z}}_4$-additive 1-perfect code gives an extended 1-perfect code. 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