# Geometry on a space of polynomial functions

I am considering some geometric concepts in a space of functions.But I am not sure the concept I consider is already defined in some references.

Let $P_{1},...,P_{N}:\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be functions with $n$ variables

$dist (P,Q):=$ the number of input strings $x$ such that $P(x)$ agrees with $Q(x)$

Let $\mathcal{P}$ be a family of funcitons $\{P_{1},P_{2},...,P_{N}\}$

A subfamily $(m,\epsilon)-\mathcal{C}$ of $\mathcal{P}$ is called $(m,\epsilon)$-colony , if $(m,\epsilon)-\mathcal{C}$ is a subfamily of $\mathcal{P}$ such that for $some$ partial assignment to $m$ variables $\rho (m):{[n] \choose m} \rightarrow \{0,1\}$ $\max\limits_{i,j}dist (P_{i}|_{\rho},P_{j}|_{\rho}) \leq \epsilon$

My question : is there a reference or results on $any$ concepts relating with the above $(m,\epsilon)$-colony ?

• This seems related to Fourier analysis of the hypercube, more than any thing that has to do with geometry. – Sariel Har-Peled Nov 9 '13 at 5:32