{"paper":{"title":"Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"quant-ph","authors_text":"Ali Assem Mahmoud","submitted_at":"2026-07-28T19:31:39Z","abstract_excerpt":"The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, $\\F_{q^2}$ over $\\F_q$, is such an alphabet. Quantum error correction, however, only ever sees \\emph{self-orthogonal} additive codes, and whether that rigidity rescues the theorem---equivalently, whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation---was asked by Gluesing-Luerssen and Pllaha and answered negatively by Pllaha for particular qubit codes. We develop the negative answer systematic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26214","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/2607.26214/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"}