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We show that for any two codes the distance in $\\Gamma(n,k)_{q}$ coincides with the distance in $\\Gamma_{k}(V)$ only in the case when $n<(q+1)^2+k-2$, i.e. if $n$ is sufficiently large then for some pairs of codes the distances in the graphs $\\Gamma_{k}(V)$ and $\\Gamma(n,k)_{q}$ are "},"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":"1506.00215","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-05-31T11:00:33Z","cross_cats_sorted":[],"title_canon_sha256":"bc552cee834721672b5c7c4f7c79bdfe2c750c7f228c8fcb107bbf74a25c38b8","abstract_canon_sha256":"08d3e9fdba98b2a21c842d298a36493440cd4d67a7200f60d27b29d58a60697f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:55.176004Z","signature_b64":"XvXJkqHSRAC28qoN5AZbTBHgXiGJSCGul6QmlkZeQVeW7yG2Ky/it0/VKDfzzlN1PCrq4sjat0j8bU/e2VxOBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad79162abbfe1d0afea8ac73a6e74ae51c7e7a05aa25b4c699d457b9e54ea3bf","last_reissued_at":"2026-05-18T01:59:55.175542Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:55.175542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the distance between linear codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mariusz Kwiatkowski, Mark Pankov","submitted_at":"2015-05-31T11:00:33Z","abstract_excerpt":"Let $V$ be an $n$-dimensional vector space over the finite field consisting of $q$ elements and let $\\Gamma_{k}(V)$ be the Grassmann graph formed by $k$-dimensional subspaces of $V$, $1<k<n-1$. Denote by $\\Gamma(n,k)_{q}$ the restriction of $\\Gamma_{k}(V)$ to the set of all non-degenerate linear $[n,k]_{q}$ codes. We show that for any two codes the distance in $\\Gamma(n,k)_{q}$ coincides with the distance in $\\Gamma_{k}(V)$ only in the case when $n<(q+1)^2+k-2$, i.e. if $n$ is sufficiently large then for some pairs of codes the distances in the graphs $\\Gamma_{k}(V)$ and $\\Gamma(n,k)_{q}$ are "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00215","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":"1506.00215","created_at":"2026-05-18T01:59:55.175629+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.00215v1","created_at":"2026-05-18T01:59:55.175629+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.00215","created_at":"2026-05-18T01:59:55.175629+00:00"},{"alias_kind":"pith_short_12","alias_value":"VV4RMKV37YOQ","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"VV4RMKV37YOQV7VI","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"VV4RMKV3","created_at":"2026-05-18T12:29:47.479230+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/VV4RMKV37YOQV7VIVRZ2NZ2K4U","json":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U.json","graph_json":"https://pith.science/api/pith-number/VV4RMKV37YOQV7VIVRZ2NZ2K4U/graph.json","events_json":"https://pith.science/api/pith-number/VV4RMKV37YOQV7VIVRZ2NZ2K4U/events.json","paper":"https://pith.science/paper/VV4RMKV3"},"agent_actions":{"view_html":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U","download_json":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U.json","view_paper":"https://pith.science/paper/VV4RMKV3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.00215&json=true","fetch_graph":"https://pith.science/api/pith-number/VV4RMKV37YOQV7VIVRZ2NZ2K4U/graph.json","fetch_events":"https://pith.science/api/pith-number/VV4RMKV37YOQV7VIVRZ2NZ2K4U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U/action/storage_attestation","attest_author":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U/action/author_attestation","sign_citation":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U/action/citation_signature","submit_replication":"https://pith.science/pith/VV4RMKV37YOQV7VIVRZ2NZ2K4U/action/replication_record"}},"created_at":"2026-05-18T01:59:55.175629+00:00","updated_at":"2026-05-18T01:59:55.175629+00:00"}