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For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show that this invariant is determined by the extended weight enumerator of the code.\\\\ The hull of a code is not invariant under monomial equivalence if $q\\geq 4$. We show that every ${\\mathbb F}_q $-linear code is monomial equivalent with an LCD code 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":"1707.08856","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-07-27T13:24:48Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"ff52e4ac21052a1f6fa3cd6e383782fa31719929e29f9813e34f7b99393a0c38","abstract_canon_sha256":"e4f4a5779b686d14e1e62019d47e4f898b6c1e7c0dfccaa8be654d0c254764f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:46.268587Z","signature_b64":"uDtDJAm7UeVkh0tfRfesNH3kfuOOdldBS+2GHb4B/LAaVvCAGMJAltv7uRju8yaGS8FX02tJzZLjjbj134IJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad4768539fe197ef3dd9a28b1d0eec349b637e9e91ae2b73c5f332e544f40bd6","last_reissued_at":"2026-05-18T00:05:46.268034Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:46.268034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"LCD codes over ${\\mathbb F}_q $ are as good as linear codes for q at least four","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Ruud Pellikaan","submitted_at":"2017-07-27T13:24:48Z","abstract_excerpt":"The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \\cap C^\\perp$. 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