Are there Similar Distance Binary Error Correcting Codes? I'm trying to find a low distortion embedding of the trivial metric space into hamming space.

It seems this should be doable by using a large set of low dimensional vectors, with approximately equal pairwise distance.

My question is if it makes sense to expect error correcting codes to have this property?

Usually when designing error correcting codes, we are interested in finding the highest achievable rate, given a minimum distance between points. A similar question is to consider, instead of the minimum distance, the ratio between the maximum and minimum distance, $\rho=max/min$. Given $\rho$, what codes should one consider for maximizing the rate?

I tried to count the distances in some of the codes from this list of optimal binary codes:


The $(7,2^3,4)$ code has all distances equal to four, so $\rho=1$. 
The $(8,2^4,4)$ code has $112$ pairs of distance $4$ and $8$ of distance $8$, so it has $\rho=2$. 
The $(9,40,3)$ code has maximum distance $8$, so $\rho=8/3$.
The $(24,2^{12},8)$ golay code has maximum distance $24$, so $\rho=3$.


The two later codes are nearly as bad as possible. Are there any codes for which I shouldn't expect this to be the case? If not, can I at least say something about the distribution of distances being fairly concentrated?
Every $\epsilon$-biased set gives a code whose minimal relative distance is $0.5 - \epsilon$ and maximal relative distance is $0.5 + \epsilon$.

To see it, write the elements of the set as the rows of a matrix, and then define the code to be the span of the columns of the matrix. The $\epsilon$-biased property of the set is equivalent to saying that the relative distance between codewords is always between $0.5 - \epsilon$ and $0.5 + \epsilon$.

One particular construction of such sets will give you a code whose dimension is linear in its block length. It is mentioned here:
http://www.wisdom.weizmann.ac.il/~benaroya/SmallBiasNew.pdf

Basically, the idea is to take AG codes of constant rate and relative distance close to $1$, and concatenating them with the Hadamard code.