Here is a list of block codes of length \(n\) that detect or correct an error on at most two coordinates.

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

Best \((10,40,4)\) code | Binary nonlinear \((10,40,4)\) code that is unique [1]. Under Construction \(A\), this code yields \(P_{10c}\), a non-lattice sphere packing that is the densest known in 10 dimensions [2][3; pg. 140]. |

Extended Hamming code | Member of an infinite family of binary linear codes with parameters \([2^r,2^r-r-1, 4]\) for \(r \geq 2\) that are extensions of the Hamming codes by a parity-check bit. |

Hamming code | Member of 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. Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code. |

Hexacode | The \([6,3,4]_4\) self-dual MDS code with generator matrix \begin{align} \begin{pmatrix} 1 & 0 & 0 & 1 & 1 & \omega\\ 0 & 1 & 0 & 1 & \omega & 1\\ 0 & 0 & 1 & \omega & 1 & 1 \end{pmatrix}~, \tag*{(1)}\end{align} where \(GF(4) = \{0,1,\omega, \bar{\omega}\}\). Has connections to projective geometry, lattices [3], and conformal field theory [4]. |

Julin-Golay code | One of several nonlinear binary \((12,144,4)\) codes based on the Steiner system \(S(5,6,12)\) [5,6][7; Sec. 2.7][8; Sec. 4] or their shortened versions, the nonlinear \((11,72,4)\), \((10,38,4)\), and \((9,20,4)\) Julin-Golay codes. Several of these codes contain more codewords than linear codes of the same length and distance and yield non-lattice sphere-packings that hold records in 9 and 11 dimensions. |

Melas code | Cyclic \([2^m -1, 2^m - 1 - 2m, 5]\) linear code with generator polynomial is \(g(x) = p(x)p(x)^{\star}\), where \(p(x)\) is a primitive polynomial of degree \(m\) that is the minimal polynomial over \(GF(2)\) of an element \(\alpha\) of order \(2^m -1\) in \(GF(2^m)\), \(m\) is odd and greater that five, and '\(\star\)' denotes reciprocation [9]. |

Nordstrom-Robinson (NR) code | A nonlinear \((16,256,6)\) binary code that is the smallest Kerdock and the smallest Preparata code. The size of this code is larger than the largest possible linear code with the same length and distance. |

Pentacode | Nonlinear \((5,40,4)_{\mathbb{Z}_4}\) code over \(\mathbb{Z}_4\) whose codewords are all cyclic permutations and negations of the strings \(01112\), \(03110\), \(21310\), and \(21132\). |

Preparata code | A nonlinear binary \((2^{m+1}-1, 2^{m+1}-2m-2, 5)\) code where \(m\) is odd. The size of this code is twice the size of the largest possible linear code with the same length and distance. |

Single parity-check (SPC) code | Also known as a sum-zero, zero-sum, or even-weight code. An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit such that the parity (i.e., sum over all coordinates of each codeword) is zero. If the Hamming weight of a message is odd (even), then the parity bit is one (zero). This code requires only one extra bit of overhead and is therefore inexpensive. |

Sloane-Whitehead code | Member of an infinite \((n,\lambda\cdot 2^{n-m-1},3)\) nonlinear code family for any \(n\) satisfying \(2^m \leq n < 3.2^{m-1}\) for some \(m\) and for \(\lambda\in\{20/16,19/16,18/16\}\). Such a code has more codewords than any linear code with the same length and distance. The code is constructed by applying the \((u|u+v)\) construction recursively to the Julin-Golay codes. |

Small-distance block code | A block code of length \(n\) that either detects or corrects errors on at most two coordinates, i.e., has distance \(d \leq 5\). |

Small-distance block quantum code | A block quantum code on \(n\) subsystems that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\). |

Ternary Golay code | A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [3] and sporadic simple groups [7]. 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 \([4,2,3]_3\) self-dual MDS code with generator matrix \begin{align} \begin{pmatrix}1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2 \end{pmatrix}~, \tag*{(2)}\end{align} where \(GF(3) = \{0,1,2\}\). Has connections to lattices [3]. |

Vasilyev code | Member of an infinite \((2^{m+1}-1,2^{2n-m},3)\) family of perfect nonlinear codes for any \(m \geq 3\). Constructed by applying a modification of the \((u|u+v)\) construction to a perfect \((2^m-1,2^{n-m},3)\) code, not necessarily linear [7; pg. 77]. |

\([7,4,3]\) 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)~. \tag*{(3)}\end{align} Up to equivalence, this is the only nontrivial length-seven perfect binary code containing the zero vector. |

\([8,4,4]\) extended Hamming code | Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly-even self-dual 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 \(r \geq 2\). |

\(q\)-ary parity-check code | Also known as a sum-zero or zero-sum code. An \([n,n-1,2]_q\) linear \(q\)-ary code whose codewords consist of the message string appended with a parity-check or zero-sum check digit such that the sum over all coordinates of each codeword is zero. |

## References

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