Abstract
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.
Highlights
Let Fq be a finite field with q elements and let X be a set of matrices over Fq
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 Fq and (ii) X is the set of symmetric matrices over Fq for odd q
The electronic journal of combinatorics 23(2) (2016), #P2.8 which is 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
Summary
Let Fq be a finite field with q elements. Let X be a set of matrices of the same size over Fq and let Xk contain all matrices in X of rank k. For the symmetric matrices we have the following result for finite fields of odd characteristic. (There is a more general definition [2] of the bent property for arbitrary functions from Fmq to Fq, which is not required here.) Recall that there is a one-to-one correspondence between quadratic forms on Fmq and m × m alternating matrices over Fq if q = 2 and m × m symmetric matrices over Fq if q is odd. Bent-negabent quadratic forms on Fm2 can only exist if m is even It has been shown in [15, Theorem 8] that a quadratic form on Fm2 is bent-negabent if and only if its associated alternating matrix M is such that M and M + I + J are both nonsingular, where I and.
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