Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties

Question

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$. What are all the possible $n$ ($\geq 6$) for which there exists a matrix $X$ of size $n \times 10$, such that any $6$ rows of $X$ jointly with any $4$ rows of $I$ form a $10 \times 10$ invertible matrix?

Answer 1

There is no such $n$. Assume that it exists, then $n=6$ also satisfies this property. Denote the rows of $X$ by $r_1,\ldots,r_6$, the columns of $X$ by $c_1,\ldots,c_{10}$, the rows of $I$ by $e_1,\ldots,e_{10}$. Assume that, say, $c_1,\ldots,c_{6}$ are linearly dependent. Then the minor formed by the first 6 rows and columns of $X$ equals to 0, this yields that $r_1,\ldots,r_6,e_7,e_8,e_9,e_{10}$ are linearly dependent. So any 6 vectors in the set $\{c_1,\ldots,c_{10}\}$ are linearly independent. But $c_1,\ldots,с_7$ are linearly dependent. Thus their linear dependence is $c_1+\ldots+c_7=0$. Analogously $c_8+c_1+c_2+\ldots+c_6=0$ and we get $c_7=c_8$, a contradiction.