Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors. Then we construct the following matrix:

$A=(a_{ij})_{2^{ct},2^{ct}}$, where $a_{ij}=1$ if $\bigvee_{1\leq l\leq t}v_{i}(l)\wedge v_{j}(l)=\emptyset$; $0$ otherwise.

I want to know an upper bounded of rank of matrix $A$. I hope this upper bound is as small as possible.

Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors. Then we construct the following matrix:

$A=(a_{ij})_{2^{ct},2^{ct}}$, where $a_{ij}=1$ if $\bigvee_{1\leq l\leq t}v_{i}(l)\wedge v_{j}(l)=\emptyset$; $0$ otherwise.

I want to know an upper bound for the rank of the matrix $A$. I hope this upper bound is as small as possible.

Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors. Then we construct the following matrix:

$A=(a_{ij})_{2^{ct},2^{ct}}$, where $a_{ij}=1$ iff $\bigvee_{1\leq l\leq t}v_{i}(l)\wedge v_{j}(l)=\emptyset$; $0$ otherwise.

I want to know an upper bounded of rank of matrix $A$. I hope this upper bound is as small as possible.

Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors. Then we construct the following matrix:

$A=(a_{ij})_{2^{ct},2^{ct}}$, where $a_{ij}=1$ if $\bigvee_{1\leq l\leq t}v_{i}(l)\wedge v_{j}(l)=\emptyset$; $0$ otherwise.

I want to know an upper bounded of rank of matrix $A$. I hope this upper bound is as small as possible.