Here is a list of codes related to combinatorial designs.

Code | Relation |
---|---|

Algebraic LDPC code | Combinatorial designs can be used to construct explicit LDPC codes [1–3]. |

Combinatorial design code | |

Cyclic linear \(q\)-ary code | The supports of all fixed-weight codewords of a \(q\)-ary cyclic code support a combinatorial \(1\)-design [4; Corr. 5.2.4]. |

Dodecacode | There exists a \(5\)-\((12, 6, 3)\) design in the dodecacode, and a \(3\)-\((11, 5, 4)\) design in the shortened dodecacode [5]. |

Dual linear code | Linear codes and their duals are related to combinatorial designs via the Assmus-Mattson theorem [6] (see [4; Sec. 5.4]). |

Extended Hamming code | Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [7; pg. 89]. |

Gallager (GL) code | Some Steiner systems can be used to construct Gallager codes [8]. |

Galois-field \(q\)-ary code | Designs can be constructed from \(q\)-ary codes by taking the supports of a subset of codewords of constant weight. |

Golay code | The supports of the weight-seven (weight-eight) codewords of the (extended) Golay code support the Steiner system \(S(4,7,23)\) (\(S(5,6,12)\)) [7; pg. 89]. Its blocks are called octads. |

Higman-Sims graph-adjacency code | Codewords of weight 36 of the Higman-Sims graph-adjacency code form a \(2\)-\((100,36,525)\) design [9; Remark 1.7] |

Hoffman-Singleton cycle code | The incidence matrix of the Hoffman-Singleton graph can be converted into a \(2\)-\((50,14,13)\) design [9; Prop. 1.1]. |

Jump code | Certain types of combinatorial designs can be used to obtain jump codes [10–12]. |

Mixed code | Combinatorial designs have been generalized to mixed alphabets [13]. |

Nordstrom-Robinson (NR) code | NR codewords give \(3\)-\((16, 6, 4)\), \(3\)-\((16, 8, 3)\), and \(3\)-\((16, 10, 24)\) designs [14; pg. 164]. |

Perfect code | Perfect codes and combinatorial designs are related [15,16]. |

Preparata code | Preparata codewords of each weight form a 3-design [14; pg. 471]. |

Reed-Muller (RM) code | Fixed-weight RM codewords of weight less than \(2^m\) support combinatorial 3-designs [4; Ex. 5.2.7]. |

Self-dual linear code | Self-dual extremal codes yield combinatorial \(\leq 5\)-designs using the Assmus-Mattson theorem [6] (see [4; Sec. 5.4]). See [17; Table 1.61, pg. 683] for a table of combinatorial designs obtained from self-dual codes. |

Spherical design code | |

Ternary Golay code | The supports of the weight-five (weight-six) codewords of the (extended) ternary Golay code support the Steiner system \(S(4,5,11)\) (\(S(5,6,12)\)) [7; pg. 89]. Its blocks are called hexads. |

\([2^r-1,2^r-r-1,3]\) Hamming code | Weight-three codewords of the \([2^r-1,2^r-r-1, 3]\) Hamming code support the Steiner system \(S(2,3,2^r-1)\) [7; pg. 89]. |

\([48,24,12]\) self-dual code | Fixed-weight codewords of extremal self-dual doubly-even codes whose length divides 24 form a combinatorial 5-design [6]. |

\([7,4,3]\) Hamming code | Weight-three and weight-four codewords of the \([7,4,3]\) Hamming code support combinatorial \(2\)-\((7,3,1)\) and \(2\)-\((7,4,2)\) designs, respectively [4; Ex. 5.2.5]. |

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