# 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, $\Pr[\mathsf{ECC}(v) \in \{0,1\}^m]>1-\epsilon$, 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 $$\Pr[\mathsf{ECC}_i(v) \in \{0,1\}]> 1-\epsilon$$ 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 $i\in[m]$.

• Out of curiosity, do you have any applications in mind? – Tsuyoshi Ito May 30 '12 at 19:30
• Yes, I actually have a couple of applications for an error-correcting-code with such property. However, I think it is infeasible to explain within the scope of a comment. You can contact me by mail if you're interested. – user887 May 31 '12 at 7:01
• Thanks for the reply. If it does not fit in a comment, I probably do not have time to understand the whole thing anyway, so I will leave it as it is. Thanks! – Tsuyoshi Ito Jun 7 '12 at 23:46