Where can I find examples of error correcting codes of the following types?
If you just need any code $E : \{0,1\}^n \to \{0,1\}^m$ where $m=O(n)$ and where the distance is linear in $m$, then what you are looking for is called an "asymptotically good code". There are many explicit constructions of such codes, and you can find the basic ones in lecture notes of courses about coding theory. For example, you can find a description of a classic construction in Lecture 7 here. Another example of a construction is expander codes, which are described in Lecture 14 there.

If you are looking specifically for codes where the distance between any two codewords is close to $\frac{m}{2}$, and in particular is upper bounded by $(1+\delta)\cdot \frac{m}{2}$, then things are a bit more complicated. Such codes are tightly related to objects called "$\epsilon$-biased sets", which have been studied for quite a while in TCS. You can find a very recent construction of such codes here. The earliest constructions can be found here and here (although they only give you $m = \rm{poly}(n)$).