  We propose a framework to study the effect of local recovery requirements of
codeword symbols on the dimension of linear codes, based on a combinatorial
proxy that we call \emph{visible rank}. The locality constraints of a linear
code are stipulated by a matrix $H$ of $\star$'s and $0$'s (which we call a
"stencil"), whose rows correspond to the local parity checks (with the
$\star$'s indicating the support of the check). The visible rank of $H$ is the
largest $r$ for which there is a $r \times r$ submatrix in $H$ with a unique
generalized diagonal of $\star$'s. The visible rank yields a field-independent
combinatorial lower bound on the rank of $H$ and thus the co-dimension of the
code.
  We prove a rank-nullity type theorem relating visible rank to the rank of an
associated construct called \emph{symmetric spanoid}, which was introduced by
Dvir, Gopi, Gu, and Wigderson~\cite{DGGW20}. Using this connection and a
construction of appropriate stencils, we answer a question posed in
\cite{DGGW20} and demonstrate that symmetric spanoid rank cannot improve the
currently best known $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the
dimension of $q$-query locally correctable codes (LCCs) of length $n$.
  We also study the $t$-Disjoint Repair Group Property ($t$-DRGP) of codes
where each codeword symbol must belong to $t$ disjoint check equations. It is
known that linear $2$-DRGP codes must have co-dimension $\Omega(\sqrt{n})$. We
show that there are stencils corresponding to $2$-DRGP with visible rank as
small as $O(\log n)$. However, we show the second tensor of any $2$-DRGP
stencil has visible rank $\Omega(n)$, thus recovering the $\Omega(\sqrt{n})$
lower bound for $2$-DRGP. For $q$-LCC, however, the $k$'th tensor power for
$k\le n^{o(1)}$ is unable to improve the $\widetilde{O}(n^{(q-2)/(q-1)})$ upper
bound on the dimension of $q$-LCCs by a polynomial factor.
