Geodesic
On quadratic form rank distribution and asymptotic bounds of affinity level
Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 36-38.

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An affinity level $\operatorname{la}(f)$ of a Boolean function $f$ is defined as minimal number of variable fixations, that produce an affine function. A general affinity level $\mathcal La(f)$ of Boolean function $f$ is defined as minimal number of fixations of variable linear combination values, that produce an affine function. The general affinity level of quadratic form equals $r$ iff the rank of quadratic form equals $2r$. The paper contains some asymptotic properties of the rank of quadratic forms. As a corollary some asymptotic bounds of the general affinity level of quadratic forms are formulated. Let $0\le c\le1$. If $n=2k+\varepsilon$, $0\le\varepsilon\le1$, then for almost all quadratic forms of $n$ variables, $$ \operatorname{la}(f)\geq\mathcal La(f)\geq k-\left\lceil\sqrt{\frac12\log_2 k}+\frac{c+(-1)^\varepsilon}2\right\rceil+1 $$ as $n\to\infty$.
Keywords: Boolean functions, affinity level, quadratic form. 
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     title = {On quadratic form rank distribution and asymptotic bounds of affinity level},
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A. V. Cheremushkin. On quadratic form rank distribution and asymptotic bounds of affinity level. Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 36-38. http://geodesic.mathdoc.fr/item/PDMA_2016_9_a14/

[1] Dixon L. E., Linear Groups with an Expositions to the Galois Field Theory, Publ. by B. G. Teubner, Leipzig, 1901

[2] Dedonne Zh., Geometriya klassicheskikh grupp, Mir, M., 1974 | MR

[3] Mak-Vilyams F. Dzh., Sloen N. Dzh. A., Teoriya kodov, ispravlyayuschikh oshibki, Svyaz, M., 1979

[4] Ryazanov B. V., Checheta S. I., “O priblizhenii bulevoi funktsii mnozhestvom sluchainykh kvadratichnykh form”, Diskretnaya matematika, 7:3 (1995), 129–145 | MR | Zbl

[5] Buryakov M. L., “Asimptoticheskie otsenki urovnya affinnosti dlya pochti vsekh bulevykh funktsii”, Diskretnaya matematika, 20:3 (2008), 73–79 | DOI | MR | Zbl