While considering a certain type of computational problems, I have encountered the following probabilistic problem over the two element field $GF_{2}$.
For $c\in \mathbb{Q}_{>0}$, let $p_{c}(n)$ be the probability for a matrix over $GF_{2}$ with $n$ columns and $c\cdot n$ rows, such that each row contains precisely $3$ non-zero entries, to have a full rank. Is there a $c\geq 1$ with $\lim_{n\rightarrow \infty} p_{c}(n)>0 $?
Im only interested in the case $c>1$; there are results of Calkin from 1997 that solve the case $c\leq 1$, see Dependent Sets of Constant Weight Binary Vectors. Calkin proved that there exists a sharp treshold $0<\beta<1$ with $\lim_{n\rightarrow \infty} p_{c}(n)=1$ for $c\in (0,\beta)$ and $\lim_{n\rightarrow \infty} p_{c}(n)=0$ for $c\in (\beta,1]$.
Now if I am not mistaken Calkin's results do not have any implications for the case $c>1$. I would be very happy to see $\lim_{n\rightarrow \infty} p_{c}(n)>0 $ for some $c>1$ and thought that some of you guys might know these kind of problems or simply see an obvious answer. I have been googling for a while without any success.
Ps.: There is a positive answer already for $c=1$ if one considers arbitrary $(c\cdot n) \times n$ matrices over $GF_{2}$. A simple counting exercise shows $p_{1}(n)=(1-1/2)\cdot ... \cdot (1-1/2^{n})$ whose limit lies somewhere in $(0,1)$ (my source is https://arxiv.org/pdf/math/0102059.pdf, p. 38). However, these behave very differently compared to the $3$-nonzero entries case.