Symmetric doubly dual hyperovals have an odd rank
type: Item
instance of: scholarly article
title: Symmetric doubly dual hyperovals have an odd rank
zbMATH Open document ID: 1320.51009
author: Ulrich Dempwolff
published in: Designs, Codes and Cryptography
publication date: 2015-01-23T00:00:00Z
review text: A set \(\mathcal S\) of \(n\)-dimensional subspaces of a finite-dimensional vector space \(U\) over \(\mathbb F_q\) is a dual hyperoval of rank \(n\) in \(U\), if \(|\mathcal S|=(q^n-1)/(q-1)+1\), any two elements of \(\mathcal S\) meet in a \(1\)-dimensional space, any three elements of \(\mathcal S\) meet in the zero space, and \(\mathcal S\) spans \(U\). In this case, \(\mathcal S\) is called doubly dual if it is also a dual hyperoval of rank \(n\) in the vector space dual to \(U\); then \(\dim U=2n\). Let \(\mathbb F_q=\mathbb F_2\) and let \(U=X\oplus Y\) with \(\dim X=n\). Then, the dual hyperoval \(\mathcal S\) is called bilinear, if there is a monomorphism \(\beta: X\to \mathrm{Hom}(X,Y)\) such that \(\mathcal S=\mathcal S_\beta:=\{S_e\mid e\in X\}\), where \(S_e:=\{(x,x\beta(e)\mid x\in X\}\). The bilinear dual hyperoval \(\mathcal S_\beta\) is symmetric if \(\beta=\beta^o:X\to \mathrm{Hom}(X,Y)\) with \(x\beta^o(e):=e\beta(x)\), and alternating if \(e\beta(e)=0\) for all \(e\in X\). Consider the special case \(Y=X\). Then, \(\mathcal S_\beta\) may be doubly dual. Fix a non-degenerate bilinear form \((\, , \,)\) on \(X\) and let \(\beta^t:X\to \mathrm{Hom}(X,X)=\mathrm{End}(X): e\mapsto \beta(e)^t\), where \(\beta(e)^t\) is the adjoint of \(\beta(e)\) with respect to \((\, , \,)\). The mappings \(\beta\mapsto\beta^o\) and \(\beta\mapsto\beta^t\) are called the Knuth operations. \textit{Y. Edel} [``On some representations of quadratic APN functions and dimensional dual hyperovals'', RIMS Kôkyûroku 1687, 118--130 (2010)] used the Knuth operations in order to prove one direction of \textit{H. Taniguchi}'s theorem [Finite Fields Appl. 15, No. 6, 673--681 (2009; Zbl 1185.51012)], which says that an alternating dual hyperoval is doubly dual if, and only if, it has odd rank. In the present paper, also the other direction of this theorem is proved using the Knuth operations. Moreover, also the following theorem is proved by applying this tool: A symmetric doubly du…
reviewed by: Andrea Blunck
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 6390336
zbMATH Keywords: bilinear dual hyperoval
zbMATH Keywords: Knuth operations
MaRDI profile type: Publication
full work available at URL: s10623-013-9847-y
OpenAlex ID: W2084360777
cites work: Alternating bilinear forms over GF(q)
cites work: Dimensional doubly dual hyperovals and bent functions
cites work: On quadratic APN functions and dimensional dual hyperovals
cites work: On the duals of certain \(d\)-dimensional dual hyperovals in \(\mathrm{PG}(2d+1,2)\)
cites work: Q5480746
DOI: 10.1007/S10623-013-9847-Y
description: scientific article; zbMATH DE number 6390336