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The aim of the present paper, is to study QC codes of length $s\\ell$ with index $s$ over the finite field $\\mathbb{F}$ and find generator polynomials and generator matrix for these codes. To achieve this aim, we apply a novel method to find generator polynomials for $\\mathbb{F}[y]$-submodules of $\\frac{\\mathbb{F}[x,y]}{< x^s-1,y^{\\ell}"},"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":"1704.08815","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-04-28T05:56:01Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"fa2e2a3062d6ab4d645170684ddb5c2b2d2a7114a110790d5515339627904dfc","abstract_canon_sha256":"9888d496d5266a4b0a55b6fd650b78f0e6ab65c639a617f07cdb57b9e53f1082"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:25.217773Z","signature_b64":"RJywAUdEz0Rccbb++ZOJy+ptKT8CuSglBWXaWBHkt7y/kmZbfLt9c8yjSjC/7dh/MQgt40BWyAtzkvUMhFl3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3dd29f5648e45f210cbe394e93caabcdaf938a58b098692e53c52ed2e7208a4","last_reissued_at":"2026-05-18T00:45:25.217367Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:25.217367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generator polynomials and generator matrix for quasi cyclic codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Zahra Sepasdar","submitted_at":"2017-04-28T05:56:01Z","abstract_excerpt":"Quasi-cyclic (QC) codes form an important generalization of cyclic codes. 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