Assume $n$ is a postive integer much larger than $\ell$. Let $\tau$ be a distribution over $\mathbb{F}_p^\ell$ satisfying $n\cdot\tau(x)\in\mathbb{N}$ for all $x\in \mathbb{F}_p^\ell$ (and $\sum_{x\in\mathbb{F}_p^{\ell}}\tau(x)=1$.) A matrix $M\in \mathbb{F}^{n\times \ell}$ is called to be in $\mathcal{M}_{n,\tau}$ if the number of $x\in\mathbb{F}^\ell$ as rows of $M$ is same as $n\tau(x)$ for all $x\in\mathbb{F}^\ell$. On the other hand, the occurrence of rows of $M\in\mathbb{F}^{n\times \ell}$ divided by $n$ also defines a distribution over $\mathbb{F}_p^{\ell}$.
For any two matrices $M_1,M_2\in\mathcal{M}_{n,\tau}$, let $M$ be the $n\times (2\ell)$ matrix such that $M=[M_1,M_2]$, i.e., its first submatrix is $M_1$ and the second submatrix is $M_2$. Then $M$ defines some distribution $\mu$ over $\mathbb{F}_p^{2\ell}$.
I am wondering is it true that the entropy $H(\mu):=-\sum_{x\in supp(\mu)}\mu(x)\log\mu(x)\le H(v)$?
If not, let $d(\tau)=\dim(span(supp(\tau)))$. Is it true that $$H(\mu)d(\tau)<H(\tau)d(\mu)$$ when $d(\mu)<2d(\tau)$?
We can treat the first question without reference to the vector space - you're effecive constructing a law on two random variables $(U,V)$ such that the marginal laws of $U$ and $V$ are both $\tau$ (up to some integrality conditions). Then $$ H(\tau) = H(U) \le H(U) + H(V|U) = H(\mu)\le H(U) + H(V) = 2H(\tau),$$ where for a random variable $X$, $H(X)$ is the entropy of its law. As long as $n\tau$ is favourible, the upper bound can be achieved - for example, if $\tau(x) \in \{0, 1/k\}$ for some $k$ that divides $n$.
For the second question, consider the case where $\tau$ is uniform on two linearly independent vectors $v_1,v_2$, and $n$ is divisible by $4$. Then by uniformly mixing up the $v_1$ rows with $v_2$ rows, we can generate a $\mu$ that is uniform on $\{(v_1, v_1), (v_1, v_2), (v_2, v_1), (v_2,v_2)\}.$ But the dimension of the space spanned by this is $3$, since $(v_1,v_2) = (v_1,v_1) + (v_2, v_2) - (v_2, v_1)$. In this case $H(\tau) = 1, H(\mu) = 2,$ $d(\tau) = 2, d(\mu) = 3,$ and $4 > 3$. This should be embeddable into quite rich $\tau$s.