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We also prove necessary conditions on $(m,n)$ for $1$-perfect codes that are linear over $Z_4$ (we call such codes additive) to exist in $D(m,n)$ graphs; for some of these parameters, we show the existence of codes. For every $m$ and $n$ satisfying $2m+n=(4^t-1)/3$ and $m \\le (4^t-5\\cdot 2^{t-1}+1)/9$, we prove the existence of $1$-perfect codes in $D(m,n)$, without the restriction t"},"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":"1407.6329","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-23T18:41:52Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"e66138c158f2248912843a72c7b0d56e5517ce55b72d3456d5dea3dde8482728","abstract_canon_sha256":"b4d1db1a53075160464254cdde5f594fc1e6332ac478ac9504675b404d9a40ce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:59.821550Z","signature_b64":"BYRAV9JHtG2h37OmMPpJ51C1PkijcFcUCSKQ/bZZa5PyFGJtGZ5THBTlVTvhwWGYRm+QxAt3mZGPZ3D0tD0nDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a30156c6cb7dfa4dc89c4c421e6d501443d89380f619adf2ef9aee2c0d07a30a","last_reissued_at":"2026-05-18T01:12:59.821202Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:59.821202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Perfect codes in Doob graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Denis Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)","submitted_at":"2014-07-23T18:41:52Z","abstract_excerpt":"We study $1$-perfect codes in Doob graphs $D(m,n)$. 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