It seems, especially in the light of @Yuichiro answer, that it makes sense for me to share here with you my original constant weight codes, which I have discovered in 1977-78 but just absolutely couldn't believe that they were not known. Only years later I got clear evidence that they were still unknown to the public (only then I posted them on pl.sci.matematyka, and I informed about them Neil Sloan and his co-maintainer of the ECC tables--both worked at Bell Labs at the time; I didn't get any feedback from them though).

My construction mostly doesn't care about the finiteness. Let $K$ be an arbitrary field, let $L$ be an arbitrary proper subfield of $K$.   Let   $P(K)\ \ P(L)$   be their projective lines (1-dim projective spaces); we may assume that   $P(L)\subset P(K)$ -- it's a harmless abuse. Let   $G\ H$   be the projective groups of   $P(K)\ P(L)$   respectively. Let   $\Gamma := \Gamma(K\ L)$   be the family of all images of   $P(L)$   under the projective maps from   $G$:

$$\Gamma := \{f(P(L)) : f\in P(K)\}$$

That's it. We may call the members of   $\Gamma$   to be circles. For every three different points   $x\ y\ z\in P(K)$   there exists **exactly one** circle which contains all three of them. When   $K$   is a **finite** field then   $\Gamma$   is the best possible (even ***perfect*** or similar terminology) constant-weight code--instead of considering the binary sequences we deal equivalently with subsets of $P(K)$, or here simply with circles.

Let's say that &nbsp; $|K|=p^n$ &nbsp; and &nbsp; $|L|=p^m$, &nbsp; where &nbsp; $p$ &nbsp; is a prime, and &nbsp; $0 < m < n$ &nbsp; are two natural numbers. Thus &nbsp; $w:=p^m+1$, while @sams' &nbsp; $n$ &nbsp; is &nbsp; $p^n+1$ &nbsp; here&nbsp; (sorry for that). The minimal distance between the codes is &nbsp; $a := 2\cdot (p^m-1)$. And that's what is important for the standard theory, while the maximal distance between the codes is &nbsp; $b:=2\cdot (p^m+1) = a+4$.

Of course
$$|\Gamma|\ \ =\ \ \frac{\binom \nu 3}{\binom \mu 3}$$
where
$$\nu := p^n+1\qquad\quad \mu=p^m+1$$

The property of circles (exactly one circle passing through any three different points) follows from the exact 3-transitivity of $G$, and the identification &nbsp; $H\subset G$.

Edit: I rushed. Of course $p^n$ must be a power of $p^m$, i.e. $n$ must be a multiple of $m$.