Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a set of matrices over $\mathbb{F}_q$. The main results of this paper are explicit expressions for the number of pairs $(A,B)$ of matrices in $X$ such that $A$ has rank $r$, $B$ has rank $s$, and $A+B$ has rank $k$ in the cases that (i) $X$ is the set of alternating matrices over $\mathbb{F}_q$ and (ii) $X$ is the set of symmetric matrices over $\mathbb{F}_q$ for odd $q$. Our motivation to study these sets comes from their relationships to quadratic forms. As one application, we obtain the number of quadratic Boolean functions that are simultaneously bent and negabent, which solves a problem due to Parker and Pott.
@article{10_37236_4072,
author = {Alexander Pott and Kai-Uwe Schmidt and Yue Zhou},
title = {Pairs of quadratic forms over finite fields},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {2},
doi = {10.37236/4072},
zbl = {1335.05229},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4072/}
}
TY - JOUR
AU - Alexander Pott
AU - Kai-Uwe Schmidt
AU - Yue Zhou
TI - Pairs of quadratic forms over finite fields
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/4072/
DO - 10.37236/4072
ID - 10_37236_4072
ER -
%0 Journal Article
%A Alexander Pott
%A Kai-Uwe Schmidt
%A Yue Zhou
%T Pairs of quadratic forms over finite fields
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/4072/
%R 10.37236/4072
%F 10_37236_4072
Alexander Pott; Kai-Uwe Schmidt; Yue Zhou. Pairs of quadratic forms over finite fields. The electronic journal of combinatorics, Tome 23 (2016) no. 2. doi: 10.37236/4072
