{"paper":{"title":"Galois Hulls of Linear Codes over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Hongwei Liu, Xu Pan","submitted_at":"2018-09-21T12:03:29Z","abstract_excerpt":"The $\\ell$-Galois hull $h_{\\ell}(C)$ of an $[n,k]$ linear code $C$ over a finite field $\\mathbb{F}_q$ is the intersection of $C$ and $C^{{\\bot}_{\\ell}}$, where $C^{\\bot_{\\ell}}$ denotes the $\\ell$-Galois dual of $C$ which introduced by Fan and Zhang (2017). The $\\ell$- Galois LCD code is a linear code $C$ with $h_{\\ell}(C) = 0$. In this paper, we show that the dimension of the $\\ell$-Galois hull of a linear code is invariant under permutation equivalence and we provide a method to calculate the dimension of the $\\ell$-Galois hull by the generator matrix of the code. Moreover, we obtain that th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08053","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":""},"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"}