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We give exact values of $LCD{[}n,k{]}$ for $1 \\le k \\le n \\le 12$.\n  We also show that $LCD[n,n-i]=2$ for any $i\\geq2$ and $n\\geq2^{i}$. Furthermore, we show that $LCD[n,k]\\leq LCD[n,k-1]$ for $k$ odd and $LCD[n,k]\\leq LCD[n,k-2]$ for $k$ even."},"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":"1701.04165","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-01-16T04:49:33Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"aa6184c2c6bb68de0f96b53f9a6844e44cb0c5b3495eda75e4f0b7f1f14125cb","abstract_canon_sha256":"184d174bd40d333f023116a9216736f6ef7aac5fb49c5e9c0ba4b69b2b03cc14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:47.637738Z","signature_b64":"C2NFrasP+WIQHDoyC+varlMjwt6OPTDenz+XFsifRSZNVErRBA/t1LfJSSx9b8BWBg07nPPCLeeqDGeLcqA8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2ca9ccc384ea909f6739218a0d464e06fd1c71037e17b234cc0529b3dbeb5fe","last_reissued_at":"2026-05-18T00:52:47.637007Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:47.637007Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Bounds on Binary LCD Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Byung-Sun Won, Jon-Lark Kim, Lucky Galvez, Nari Lee, Young Gun Roe","submitted_at":"2017-01-16T04:49:33Z","abstract_excerpt":"A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\\perp}$ satisfies $C \\cap C^{\\perp}$= $\\left\\{ \\mathbf{0}\\right\\} $. Let $LCD{[}n,k{]}$ denote the maximum of possible values of $d$ among $[n,k,d]$ binary LCD codes. We give exact values of $LCD{[}n,k{]}$ for $1 \\le k \\le n \\le 12$.\n  We also show that $LCD[n,n-i]=2$ for any $i\\geq2$ and $n\\geq2^{i}$. Furthermore, we show that $LCD[n,k]\\leq LCD[n,k-1]$ for $k$ odd and $LCD[n,k]\\leq LCD[n,k-2]$ for $k$ even."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04165","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":"1701.04165","created_at":"2026-05-18T00:52:47.637126+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.04165v1","created_at":"2026-05-18T00:52:47.637126+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04165","created_at":"2026-05-18T00:52:47.637126+00:00"},{"alias_kind":"pith_short_12","alias_value":"ULFJZTBYJ2UQ","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_16","alias_value":"ULFJZTBYJ2UQT5TT","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_8","alias_value":"ULFJZTBY","created_at":"2026-05-18T12:31:46.661854+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/ULFJZTBYJ2UQT5TTSIMKBVDE4B","json":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B.json","graph_json":"https://pith.science/api/pith-number/ULFJZTBYJ2UQT5TTSIMKBVDE4B/graph.json","events_json":"https://pith.science/api/pith-number/ULFJZTBYJ2UQT5TTSIMKBVDE4B/events.json","paper":"https://pith.science/paper/ULFJZTBY"},"agent_actions":{"view_html":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B","download_json":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B.json","view_paper":"https://pith.science/paper/ULFJZTBY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.04165&json=true","fetch_graph":"https://pith.science/api/pith-number/ULFJZTBYJ2UQT5TTSIMKBVDE4B/graph.json","fetch_events":"https://pith.science/api/pith-number/ULFJZTBYJ2UQT5TTSIMKBVDE4B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B/action/storage_attestation","attest_author":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B/action/author_attestation","sign_citation":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B/action/citation_signature","submit_replication":"https://pith.science/pith/ULFJZTBYJ2UQT5TTSIMKBVDE4B/action/replication_record"}},"created_at":"2026-05-18T00:52:47.637126+00:00","updated_at":"2026-05-18T00:52:47.637126+00:00"}