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 . Cousins: Dual linear code, Ternary Golay Code.

## References


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

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

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.

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

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999). DOI

E. M. Rains and N. J. A. Sloane, “Self-Dual Codes”. math/0208001

C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948). DOI

R. W. Hamming, “Error Detecting and Error Correcting Codes”, Bell System Technical Journal 29, 147 (1950). DOI

M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.