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Left ideals of the group ring $R[D_{2n}]$ are called left dihedral codes over $R$ of length $2n$, and abbreviated as left $D_{2n}$-codes over $R$. Let ${\\rm gcd}(n,p)=1$ in this paper. Then any left $D_{2n}$-code over $R$ is uniquely decomposed into a direct sum of concatenated codes with inner codes ${\\cal A}_i$ and outer codes $C_i$, where ${\\cal A}_i$ is a cyclic code over $R$ of length $n$ and $C_i$"},"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":"1609.04083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-09-14T00:54:06Z","cross_cats_sorted":["math.IT","math.RA"],"title_canon_sha256":"c6ad0ccbd28c3d25ca884f9fabace651566d74b16ef0a8ded29fa83ec1be952a","abstract_canon_sha256":"ff37d53e710c5b985dbff09c891164f90fbadc756ae9482cea70386afa04a9c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:40.709816Z","signature_b64":"B0fIXBl0BsO8Swjaje+IFsf0AaMFC+v1AU+wni9BUc8pMeZwPfqOiQ6DMxomxq6Qvjfonjv5xjYGHZGR+wjyBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f08aa3849d85f5688eb65bff9c3c6524d6d13b4b5add2df40c5cc467adcee1db","last_reissued_at":"2026-05-18T01:04:40.709210Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:40.709210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Left dihedral codes over Galois rings ${\\rm GR}(p^2,m)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.RA"],"primary_cat":"cs.IT","authors_text":"Fang-Wei Fu, Yonglin Cao, Yuan Cao","submitted_at":"2016-09-14T00:54:06Z","abstract_excerpt":"Let $D_{2n}=\\langle x,y\\mid x^n=1, y^2=1, yxy=x^{-1}\\rangle$ be a dihedral group, and $R={\\rm GR}(p^2,m)$ be a Galois ring of characteristic $p^2$ and cardinality $p^{2m}$ where $p$ is a prime. Left ideals of the group ring $R[D_{2n}]$ are called left dihedral codes over $R$ of length $2n$, and abbreviated as left $D_{2n}$-codes over $R$. Let ${\\rm gcd}(n,p)=1$ in this paper. Then any left $D_{2n}$-code over $R$ is uniquely decomposed into a direct sum of concatenated codes with inner codes ${\\cal A}_i$ and outer codes $C_i$, where ${\\cal A}_i$ is a cyclic code over $R$ of length $n$ and $C_i$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04083","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.04083","created_at":"2026-05-18T01:04:40.709295+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.04083v1","created_at":"2026-05-18T01:04:40.709295+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04083","created_at":"2026-05-18T01:04:40.709295+00:00"},{"alias_kind":"pith_short_12","alias_value":"6CFKHBE5QX2W","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"6CFKHBE5QX2WRDVW","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"6CFKHBE5","created_at":"2026-05-18T12:30:01.593930+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET","json":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET.json","graph_json":"https://pith.science/api/pith-number/6CFKHBE5QX2WRDVWLP7ZYPDFET/graph.json","events_json":"https://pith.science/api/pith-number/6CFKHBE5QX2WRDVWLP7ZYPDFET/events.json","paper":"https://pith.science/paper/6CFKHBE5"},"agent_actions":{"view_html":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET","download_json":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET.json","view_paper":"https://pith.science/paper/6CFKHBE5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.04083&json=true","fetch_graph":"https://pith.science/api/pith-number/6CFKHBE5QX2WRDVWLP7ZYPDFET/graph.json","fetch_events":"https://pith.science/api/pith-number/6CFKHBE5QX2WRDVWLP7ZYPDFET/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET/action/storage_attestation","attest_author":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET/action/author_attestation","sign_citation":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET/action/citation_signature","submit_replication":"https://pith.science/pith/6CFKHBE5QX2WRDVWLP7ZYPDFET/action/replication_record"}},"created_at":"2026-05-18T01:04:40.709295+00:00","updated_at":"2026-05-18T01:04:40.709295+00:00"}