Here is a list of codes whose codewords have constant weight.
Code Description
Balanced code An even-length-\(n\) \(q\)-ary code whose nonzero codewords all have a Hamming weight of \(n/2\). A code is \(\epsilon\)-balanced if the relative weight (i.e., weight divided by \(n\)) of all nonzero codewords lies in the interval \([\frac{1-\epsilon}{2},\frac{1+\epsilon}{2}]\). A code is \(\gamma\)-unbiased if the relative weight lies in the interval \((\frac{1}{2}-\frac{1}{n^{\gamma}},\frac{1}{2}+\frac{1}{n^{\gamma}})\).
Combinatorial design A constant-weight binary code that is mapped into a combinatorial \(t\)-design.
Constant-weight code A block code over a field or a ring whose codewords all have the same Hamming weight \(w\). The complement of a binary constant-weight code is a constant-weight code obtained by interchanging zeroes and ones in the codewords. The set of all binary codewords of length \(n\) forms the Johnson space \(J(n,w)\) [1–4].
Hadamard code An \([2^m,m,2^{m-1}]\) balanced binary code. The \([2^m,m+1,2^{m-1}]\) augmented Hadamard code is the first-order RM code (a.k.a. RM\((1,m)\)), while the \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)).
One-hot code A length-\(n\) binary code whose codewords are those with Hamming weight one. The reverse of this code, where all codewords have Hamming weight \(n-1\) is called a one-cold code.
One-versus-one (OVO) code A length-\(n\) ternary code over \(\{\om 1,0\}\) whose whose generator matrix has columns with one \(+1\), one \(-1\), and with the rest of the entries zero.
Tetracode The \([4,2,3]_3\) self-dual MDS code that has connections to lattices [5].
Weight-two code A length-\(n\) binary code whose codewords all have Hamming weight two. Such codes provide slightly extra redundancy for storage of small-scale information such as ZIP codes or decimal digits.
\([2^m-1,m,2^{m-1}]\) simplex code A member of the code family that is dual to the \([2^m,2^m-m-1,3]\) Hamming family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping.
\([7,3,4]\) simplex code Second-smallest member of the simplex code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping.
\(q\)-ary simplex code An \([n,m,q^{m-1}]_q\) projective code with \(n=\frac{q^m-1}{q-1}\), denoted as \(S(q,m)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,q)\), with each column being a chosen representative of the corresponding element.

References

[1]
Delsarte, Philippe. "An algebraic approach to the association schemes of coding theory." Philips Res. Rep. Suppl. 10 (1973): vi+-97.
[2]
P. Delsarte, “Association schemes and t-designs in regular semilattices”, Journal of Combinatorial Theory, Series A 20, 230 (1976) DOI
[3]
Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI
[4]
V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces”, Acta Applicandae Mathematicae 29, 1 (1992) DOI
[5]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI