Boolean error correcting code over $\mathbb{F}_q$ Is there any known construction of a linear error correcting code $\mathsf{ECC}:\mathbb{F}_q^n \to \mathbb{F}_q^m$ (with reasonable parameters), such that when given a Boolean vector $v\in \{0,1\}^n$, it also returns a Boolean vector w.h.p.? (although it is over $\mathbb{F}_q$)

(that is, $\Pr[\mathsf{ECC}(v) \in \{0,1\}^m]&gt;1-\epsilon$, where the probability is taken over uniformly choosing $v\in \{0,1\}^n$, and $\epsilon$ is arbitrarily small)

If not, what if we relax the condition to
$$\Pr[\mathsf{ECC}_i(v) \in \{0,1\}]&gt; 1-\epsilon$$
Where $\mathsf{ECC}_i$ returns the $i$'th coordinate of $\mathsf{ECC}$, $\epsilon$ is arbitrarily small, and the probability is taken both over uniformly choosing $v\in \{0,1\}^n$ and uniformly choosing a coordinate $i\in[m]$.
Yes. For example, a Reed-Solomon code contains a BCH code, which is a binary linear code, as a sub-code. These are called subfield-subcodes.