Code | Description |
---|---|

Golay code | A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [1] and sporadic simple groups [2]. Adding a parity bit to the code results in the \([24, 12, 8]\) extended Golay code. Up to equivalence, both codes are unique for their respective parameters. |

Graph homology code | This code's properties are derived from the size two chain complex associated with a particular graph. Given a connected simplicial (no self loops or muliedges) graph \(G = (V, E)\), which is not a tree, with incidence matrix \(\Gamma\) we can construct a code by choosing a parity check matrix which consists of all the linearly independent rows of \(\Gamma\). This is a \([n,k,d]\) code with \(n = |E|\), \(k = 1 - \mathcal{X}(G) = 1-|V|+|E|\), where \( \mathcal{X}(G)\) is the euler characteristic of the graph. The code distance is equal to the shortest size of a cycle, guaranteed to exist since \(G\) is not a tree. |

Hamming 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. |

Perfect code | An \((n,K,2t+1)_q\) binary or \(q\)-ary code is perfect if parameters \(n\), \(K\), \(t\), and \(q\) are such that the Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t}(q-1)^{j}{n \choose j}\leq q^{n}/K \end{align} becomes an equality. For example, for a binary \(q=2\) code with one logical bit (\(K=2\)) and \(t=1\), the bound becomes \(n+1 \leq 2^{n-1}\). Perfect codes are those for which balls of Hamming radius \(t\) exactly fill the space of all \(n\) \(q\)-ary strings. |

Perfect quantum code | A non-degenerate code constructed out of \(q\)-dimensional qudits and having parameters \(((n,K,2t+1))\) is perfect if \(n\), \(K\), \(t\), and \(q\) are such that the quantum Hamming bound \begin{align} \sum_{j=0}^{t}(q^2-1)^{j}{n \choose j}\leq q^{n}/K \end{align} becomes an equality. For example, for a qubit \(q=2\) code with one logical qubit (\(K=2\)) and \(t=1\), the bound becomes \(3n+1 \leq 2^{n-1}\). The bound can be saturated only at certain \(n\). |

Repetition code | \([n,1,n]\) binary linear code encoding one bit of information into an \(n\)-bit string. The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information. The idea is to increase the code distance by repeating the logical information several times. It is a \((n,1)\)-Hamming code. |

Ternary Golay Code | A \([11,6,5]_{GF(3)}\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [1] and sporadic simple groups [2]. Adding a parity bit to the code results in the \([12, 6, 6]\) extended ternary Golay code. Up to equivalence, both codes are unique for their respective parameters. |

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 [1]. |

\([7,4,3]\) Hamming code | Second-smallest member of the Hamming code family with generator matrix \begin{align} \left(\begin{array}{ccccccccccc} 1 & 0 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{array}\right)~. \end{align} Can be extended to yield the \([8,4,4]\) extended Hamming code. |

\(q\)-ary Hamming code | Member of an infinite family of perfect linear \(q\)-ary codes with parameters \([(q^r-1)/(q-1),(q^r-1)/(q-1)-r, 3]_q\) for \(r \geq 2\). |

## References

- [1]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999). DOI
- [2]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.