# Quadratic forms and 0-1 points.

I have a quadratic form $Q(u) = \langle Du , u \rangle = 0$, where $D$ is circulant-symmetric from $\mathbb{R}^{n \times n}$ $D$ has all entries $0$ or $1$ except the diagonal which a negative real number $a$. It is known there are solutions to $Q(u)$ such that $u$ is just from $\{0,1\}^{n}$. I have a point $v \in \mathbb{R}^{n}$ on the quadratic form. How can I find an estimate of the distance to the closest point $w$ from $v$ such that the point $w$ satisfies $Q(w) = 0$ and has coordinates just $0$ or $1$? Are there any connections to lattice decoding? (Will simple rounding work?)

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A point $v$ on what? – Will Jagy Dec 26 '11 at 6:25
@Will Jagy: Hope my question makes more sense now! – Turbo Dec 26 '11 at 6:29
No, it really does not. The quadratic form is the function $Q.$ The null cone is the set of vectors satisfying $Q(v) = 0.$ There is no guarantee that there is any point $w$ with all coordinates $0,1$ that also lies on the null cone. Evidently $D$ is just a symmetric matrix. – Will Jagy Dec 26 '11 at 6:38
@Will: I am parsing this a saying that he knows for some reason that there is a (0, 1) point on the light cone... – Igor Rivin Dec 26 '11 at 12:00
@Igor That is correct! – Turbo Dec 26 '11 at 17:00