Does it make sense to have an error correction code which acts differently on different states (for example if we run something which runs on the binary string from $0^n \rightarrow 1^n$ involving all combinations, can we make it act biased towards the 1 states and not towards the 0 states)?
I'm not sure that I understood your intention correctly. Typing this as an answer as it is too long for a comment.
This is easy, if we use a light to moderate memory convolutional code. It isn't at all difficult to make Viterbi decoding algorithm interpret the received symbols asymmetrically. For example as follows: a received $0$ is treated as something that truly is a $0$ with probability $p>1/2$, and a $1$ with probability $1-p$. We can also declare that a received $1$ truly is a $1$ with probability $q$ and, $0$ with probability $1-q$. Viterbi algorithm then gives you a maximum likelihood input sequence.
If $q>p$, then the algorithm treats $1$s as more certain than $0$s. Is this the kind of bias you had in mind?
A catch is that not all error-correcting codes are amenable to Viterbi decoding. With most convolutional codes this would not pose a problem. What we really need is a trellis representation of the code. In an earlier answer I try to describe, how to construct a trellis representation of a linear block code. For the best (in terms of minimum Hamming distance) known linear codes the resulting trellises tend to have a prohibitive complexity.
Most error correcting codes are designed for something like the Binary Symmetric Channel where there is equal chance of a 0 switching to a 1 or a 1 switching to a 0.
However, information theorists have considered channels that are not symmetric in this sense. I have been told that systems that are more likely to switch from a 1 to 0 than the other way around include flash memory and some optical communication systems. Some keywords to look up are "binary asymmetric channel" or "Z-channel".
If you search google or google scholar for "coding for asymmetric channel" you'll find a number of papers. Unfortunately I don't know anything about the techniques used.