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Let \\( h_q(n) \\) denote the maximum possible codimension of such a subspace. When \\(\\gcd(q,n)=1\\), we derive necessary and sufficient conditions for \\(h_q(n)=0\\) via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of \\(\\mathbb{F}_q^n\\). We also characterize codimension-$k$ cyclically covering subspaces.\n  Based "},"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":"2606.13307","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-11T13:04:23Z","cross_cats_sorted":[],"title_canon_sha256":"32cfca59e0be840bddfdb55ac579ea9b3e1a04cbb270b30d5c47153ef1ecb959","abstract_canon_sha256":"de0fc78f52630fc3eb63126d406effd4aebb012afcf22ba08c51ff93c8f96431"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:51.541361Z","signature_b64":"/xoyUKrgpV8Y7oIdeVjf4q2Dr0MRcBYVqw39KWxj9/1GQCh6+8DaGAtL3+8NbaN2jBvWZor0oKJUZv0WsA9UDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e562630d32ba213d0455481ebe97e9a4f35115852af5659c00502eb3e3937485","last_reissued_at":"2026-06-12T01:09:51.539618Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:51.539618Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrete Fourier Transform Approach to Cyclically Covering Subspaces of $\\mathbb{F}^n_q$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pingzhi Yuan, Yangcheng Li","submitted_at":"2026-06-11T13:04:23Z","abstract_excerpt":"Let $q$ be a prime power and $n$ a positive integer. A subspace \\( U \\subseteq \\mathbb{F}_q^n \\) is called cyclically covering if the union of all its cyclic shifts covers the whole space \\( \\mathbb{F}_q^n \\). Let \\( h_q(n) \\) denote the maximum possible codimension of such a subspace. When \\(\\gcd(q,n)=1\\), we derive necessary and sufficient conditions for \\(h_q(n)=0\\) via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of \\(\\mathbb{F}_q^n\\). 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