Rank of a matrix of binomial coefficients This question arose as a side computation on error correcting codes.
Let $k$, $r$ be positive integers such that $2k-1 \leqslant r$ and let $p$ a prime number such that $r < p$. I would like to find the rank of the following $k \times k$ matrix with coefficients in the field $F_p$ 
$$
\begin{pmatrix}
\binom {r}{k} & \binom {r}{k+1} & \dotsm & \binom {r}{2k-1}\\
\vdots & \\
\binom {r}{2} & \binom {r}{3} & \dotsm & \binom {r}{k+1} \\
\binom {r}{1} & \binom {r}{2} & \dotsm & \binom {r}{k} 
\end{pmatrix}
$$
where all binomial coefficients are taken modulo $p$. I conjecture the rank should be $k$ but I have no formal proof. I am aware of Lucas's theorem, but it didn't help so far.

Write $A(r,k)$ for the matrix you describe. Then
$$
\det A(r,k) =\prod_{i,j=1}^k \frac{r-k-1+i+j}{i+j-1}.
$$
This follows from equation (2.17) in Krattenthaler's Advanced determinant calculus (with $a=n=k$, $b=r-k$). For $p\geq r+k$ every factor in the product has numerator coprime to $p$, so the reduction of $A(r,k)$ has full rank.
