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How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:

1. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?

2. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}\backslash\{0\}$?

at $b\in\Bbb Z$?

If we have $\pm1$ square matrices I believe only $2^{O(r)}\times 2^{O(r)}$ is achievable and so is upper bound and achievability both $(2b+1)^{O(r)}\times (2b+1)^{O(r)}$ here?

How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:

1. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?

2. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}\backslash\{0\}$?

at $b\in\Bbb Z$?

If we have $\pm1$ square matrices I believe only $2^{O(r)}\times 2^{O(r)}$ is achievable and so is upper bound and achievability $(2b+1)^{c_{1}r}\times (2b+1)^{c_1r}$ and $(2b+1)^{c_2r}\times (2b+1)^{c_2r}$ respectively here at some $c_1,c_2>0$?

How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:

1. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?

2. have at most $2b+1$ distinct integers?

at $b\in\Bbb Z$?

How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:

1. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?

2. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}\backslash\{0\}$?

at $b\in\Bbb Z$?

If we have $\pm1$ square matrices I believe only $2^{O(r)}\times 2^{O(r)}$ is achievable and so is upper bound and achievability both $(2b+1)^{O(r)}\times (2b+1)^{O(r)}$ here?

1. How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?

2. How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries have at most $2b+1$ distinct integers?

How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:

1. are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?

2. have at most $2b+1$ distinct integers?

at $b\in\Bbb Z$?