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These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if $d\\geqslant q+2$ then there is no generalised Reed-Solomon $[n,n-d+1,d]_{q^2}$ code which contains its Hermitian dual. We also construct an $ [\\![ 18,0,10 ]\\!] _5$ quantum MDS code, an $ [\\![ 18,0,10 ]\\!] _7"},"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":"1907.04391","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2019-07-09T20:17:38Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"2c9402e7041bb87d93fe5f1b551559e37abb3c63fe49a91a239ff54d8dfd5a67","abstract_canon_sha256":"9b2f1a1cee66b8145b9eae0b536fc468e3bae46bc74fcfe42811ca9f58293008"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:06:47.648332Z","signature_b64":"honQpgXfA9Mg+KjzW2cofxGRpxAM6OL//yT6jQjZN5lDiYlAEP4nIuaaM+WLPxBu2QoRL4OJPiHTiYlsNd4PBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"785e73881a330fca3119fa062a6da96efeee99d4f81f7ec1a3a1dfb6ed2f16f4","last_reissued_at":"2026-07-05T02:06:47.647902Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:06:47.647902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some constructions of quantum MDS codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"quant-ph","authors_text":"Simeon Ball","submitted_at":"2019-07-09T20:17:38Z","abstract_excerpt":"We construct quantum MDS codes with parameters $ [\\![ q^2+1,q^2+3-2d,d ]\\!] _q$ for all $d \\leqslant q+1$, $d \\neq q$. These codes are shown to exist by proving that there are classical generalised Reed-Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if $d\\geqslant q+2$ then there is no generalised Reed-Solomon $[n,n-d+1,d]_{q^2}$ code which contains its Hermitian dual. 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