{"paper":{"title":"Switchings of semifield multiplications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ferruh \\\"Ozbudak, Xiang-dong Hou, Yue Zhou","submitted_at":"2014-06-04T15:04:19Z","abstract_excerpt":"Let $B(X,Y)$ be a polynomial over $\\mathbb{F}_{q^n}$ which defines an $\\mathbb{F}_q$-bilinear form on the vector space $\\mathbb{F}_{q^n}$, and let $\\xi$ be a nonzero element in $\\mathbb{F}_{q^n}$. In this paper, we consider for which $B(X,Y)$, the binary operation $xy+B(x,y)\\xi$ defines a (pre)semifield multiplication on $\\mathbb{F}_{q^n}$. We prove that this question is equivalent to finding $q$-linearized polynomials $L(X)\\in\\mathbb{F}_{q^n}[X]$ such that $Tr_{q^n/q}(L(x)/x)\\neq 0$ for all $x\\in\\mathbb{F}_{q^n}^*$. For $n\\le 4$, we present several families of $L(X)$ and we investigate the de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.1067","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"}