{"paper":{"title":"Generalized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","quant-ph"],"primary_cat":"cs.IT","authors_text":"Martianus Frederic Ezerman, San Ling, Shitao Li, Yang Li, Zhonghua Sun","submitted_at":"2026-07-02T13:41:14Z","abstract_excerpt":"We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of $\\C^{\\perp_{\\rm H}}$. Every $[n+1,k+1]_{q^2}$ linear code $\\D$ with $d(\\D^{\\perp_{\\rm H}})>1$ is monomially equivalent to the generalized extended code $\\C({\\bf u},a)$ of an $[n,k]_{q^2}$ linear code $\\C$ for a fixed $a\\in\\F_{q^2}^{*}$ and some ${\\bf u}\\in\\F_{q^2}^{n}$. We then characterize the Hermitian hull and Hermitian dual distance of $\\C({\\bf u},a)$ in terms of the position of ${\\bf u}$ relative to $\\C+\\C^{\\perp_{\\rm H}}$ and t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02170","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02170/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}