Let $S_1\subset Z_2^{n_1}$ and $S_2\subset Z_2^{n_2}$, $S=(S_2,S_1)\subset Z_2^{n_1+n_2}$, I'd like to restrict $S_1$ and $S_2$ s.t The following relations hold:

$\Sigma_{1 \leq i \leq n_1+n_2}(-1)^{x_i}=0,$

$\Sigma_{1 \leq i,j \leq n_1+n_2,i\neq j}(-1)^{x_i+x_j}=0,$

$‎\vdots$‎

$\Sigma_{1 \leq i_1,i_2,...,i_l \leq n_1+n_2}(-1)^{x_{i_1}+x_{i_2}...+x_{i_l}}=0, $ for some $1 \leq l \leq n_1+n_2$,

Does anyone have any idea or seen any references? I should say it's about Boolean functions?

Let $Z_2=\{0,1\}$, $Z_2^r=Z_2\times Z_2 \times...\times Z_2,$ $S_1\subset Z_2^{n_1}$ and $S_2\subset Z_2^{n_2}$, $S=(S_2,S_1)\subset Z_2^{n_1+n_2}$, I'd like to construct $S_1$ and $S_2$ s.t the following equations hold for $S$:

For any $x=(x_{n_1+n_2},...,x_1)\in Z_2^{n_1+n_2}$

1. $\Sigma_{1 \leq i \leq n_1+n_2}(-1)^{x_i}=0,$

2. $\Sigma_{1 \leq i,j \leq n_1+n_2,i\neq j}(-1)^{x_i+x_j}=0,$

$‎\vdots$‎

l) $\Sigma_{1 \leq i_1,i_2,...,i_l \leq n_1+n_2}(-1)^{x_{i_1}+x_{i_2}...+x_{i_l}}=0,$ for some $1 \leq l \leq n_1+n_2$,

The sum is "+"over real feild,like "0+1+0+0+1+1+1=4" How should I form $S_1$ and $S_2$ s.t above equations hold? Does anyone have any idea or seen any references? I should say it's about Boolean functions.