Welcome to the Ring Kingdom.

Ring code Encodes $$K$$ states (codewords) in $$n$$ coordinates over a ring $$R$$. Parents: Error-correcting code (ECC). Parent of: $$R$$-linear code.
$$R$$-linear code A code of length $$n$$ over a ring $$R$$ is $$R$$-linear if it is a submodule of $$R^n$$. Parents: Ring code, Linear code. Parent of: Quaternary code over $$\mathbb{Z}_4$$.
Quaternary code over $$\mathbb{Z}_4$$ A linear code encoding $$K$$ states (codewords) in $$n$$ coordinates over the ring $$\mathbb{Z}_4$$ of integers modulo 4. Parents: $$R$$-linear code. Parent of: Octacode, Reed-Muller (RM) code.
Member of the RM$$(r,m)$$ family of linear binary codes derived from multivariate polynomials. The code parameters are $$[2^m,\sum_{j=0}^{r} {m \choose j},2^{m-r}]$$, where $$r$$ is the order of the code satisfying $$0\leq r\leq m$$. Punctured RM codes RM$$^*(r,m)$$ are obtained from RM codes by deleting one or more coordinates from each codeword. Parent of: Hamming code.
The unique self-dual linear code of length 8 over $$\mathbb{Z}_4$$ with generator matrix \begin{align} \begin{pmatrix} 3 & 3 & 2 & 3 & 1 & 0 & 0 & 0\\ 3 & 0 & 3 & 2 & 3 & 1 & 0 & 0\\ 3 & 0 & 0 & 3 & 2 & 3 & 1 & 0\\ 3 & 0 & 0 & 0 & 3 & 2 & 3 & 1 \end{pmatrix}\,. \end{align} Cousins: Cyclic code, Hamming code, Dual additive code.
An infinite family of perfect linear codes with parameters $$(2^r-1,2^r-r-1, 3)$$ for $$r \geq 2$$. Their $$r \times (2^r-1)$$ parity check matrix $$H$$ has all possible non-zero $$r$$-bit strings as its columns. Protection: Can detect 1-bit and 2-bit errors, and can correct 1-bit errors. Parent of: Tetracode.
The $$[4,2,3]_{GF(3)}$$ self-dual MDS code with generator matrix \begin{align} \begin{pmatrix}1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2 \end{pmatrix}~, \end{align} where $$GF(3) = \{0,1,2\}$$. Has connections to lattices [5]. Cousins: Dual linear code, Ternary Golay Code.

## References

[1]
D. E. Muller, “Application of Boolean algebra to switching circuit design and to error detection”, Transactions of the I.R.E. Professional Group on Electronic Computers EC-3, 6 (1954). DOI
[2]
I. Reed, “A class of multiple-error-correcting codes and the decoding scheme”, Transactions of the IRE Professional Group on Information Theory 4, 38 (1954). DOI
[3]
N. Mitani, On the transmission of numbers in a sequential computer, delivered at the National Convention of the Inst. of Elect. Engineers of Japan, November 1951.
[4]
J. H. Conway and N. J. A. Sloane, “Self-dual codes over the integers modulo 4”, Journal of Combinatorial Theory, Series A 62, 30 (1993). DOI
[5]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999). DOI
[6]
E. M. Rains and N. J. A. Sloane, “Self-Dual Codes”. math/0208001
[7]
C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948). DOI
[8]
R. W. Hamming, “Error Detecting and Error Correcting Codes”, Bell System Technical Journal 29, 147 (1950). DOI
[9]
M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.