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.
Reed-Muller (RM) code[1][2][3] 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. Parents: Polynomial evaluation code, Linear binary code, Quaternary code over \(\mathbb{Z}_4\), Divisible code, Group code. Parent of: Hamming code. Cousins: Binary BCH code, Dual linear code, Binary duadic code, Cyclic linear binary code, Parity-check code. Cousin of: Generalized RM (GRM) code, Hadamard code, Majorana stabilizer code, Polar code, Quantum Reed-Muller code, Quantum divisible code, Simplex code.
Octacode[4][5][6] 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} Parents: Quaternary code over \(\mathbb{Z}_4\). Cousins: Cyclic code, Hamming code, Dual additive code.
Hamming code[7][8][9] 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. Parents: Perfect code, Reed-Muller (RM) code, Binary BCH code. Parent of: Tetracode. Cousins: Projective geometry code, Binary quadratic-residue (QR) code, \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code, \(q\)-ary Hamming code, Nearly perfect code. Cousin of: Hadamard code, Octacode, Repetition code, Steane \([[7,1,3]]\) code, \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code.
Tetracode[5] 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]. Parents: Simplex code, Hamming code, Maximum distance separable (MDS) code. Cousins: Dual linear code, Ternary Golay Code.


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