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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 $(2b+1)^{c_{1}r}\times (2b+1)^{c_1r}$ and $(2b+1)^{c_2r}\times (2b+1)^{c_2r}$ respectively here at some $c_1,c_2>0$?

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  • $\begingroup$ Why $2b + 1$? In the $\pm 1$-matrices case this would evaluate to $3$ rather than $2$, right? $\endgroup$
    – Vincent
    Commented Jan 19, 2018 at 10:47
  • $\begingroup$ A lower bound is achievable by looking at finite fields of order $p=2b+1$, when $p$ is prime. Is the bound independent of the size of the matrix $n$? $\endgroup$
    – Josiah Park
    Commented Jan 21, 2018 at 1:51
  • $\begingroup$ The size of the matrix $n$ may show up in the bound unlike the $2^{O(r)}\times 2^{O(r)}$ bound. $\endgroup$
    – Josiah Park
    Commented Jan 21, 2018 at 1:57

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