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]. |

Binary BCH code | A family of BCH codes supports an infinite family of combinatorial 4-designs [4,5]. |

Combinatorial design | |

Cyclic linear \(q\)-ary code | Two families of cyclic \(q\)-ary codes support an infinite family of combinatorial 3-designs [6]. |

Dodecacode | There exists a \(5\)-\((12, 6, 3)\) design in the dodecacode, and a \(3\)-\((11, 5, 4)\) design in the shortened dodecacode [7]. |

Dual linear code | Linear codes and their duals are related to combinatorial designs via the Assmus-Mattson theorem [8,9] (see [10; Sec. 5.4]). |

EA combinatorial-design QLDPC code | Combinatorial designs can be used to construct EA QLDPC codes [11]. |

Editing code | Perfect deletion correcting codes can be constructed using combinatorial design theory [12,13]. |

Gallager (GL) code | Some Steiner systems can be used to construct Gallager codes [14]. |

Golay code | The supports of the weight-seven (weight-eight) codewords of the (extended) Golay code support the Steiner system \(S(4,7,23)\) (\(S(5,6,12)\)) [15,16][17; pg. 89]. Its blocks are called octads. |

Hadamard code | Hadamard designs are combinatorial designs constructed from Hadamard matrices [18]; see Ref. [19]. |

Higman-Sims graph-adjacency code | Codewords of weight 36 of the Higman-Sims graph-adjacency code form a \(2\)-\((100,36,525)\) design [20; 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 [20; Prop. 1.1]. |

Jump code | Certain types of combinatorial designs can be used to obtain jump codes [21–23]. |

Lexicographic code | Some lexicodes yield Steiner systems [24]. |

Mixed code | Combinatorial designs have been generalized to mixed alphabets [25]. |

Nordstrom-Robinson (NR) code | NR codewords give \(3\)-\((16, 6, 4)\), \(3\)-\((16, 8, 3)\), and \(3\)-\((16, 10, 24)\) designs [26; pg. 164]. |

Perfect code | Perfect codes and combinatorial designs are related [27,28]. |

Perfect-tensor code | Combinatorial designs and \(d\)-uniform quantum states are related [29]. |

Pless symmetry code | The supports of fixed-weight codewords of certain Pless symmetry codes support combinatorial designs [16,30,31]. |

Preparata code | Preparata codewords of each weight form a 3-design [26; pg. 471]. |

Reed-Muller (RM) code | Fixed-weight RM codewords of weight less than \(2^m\) support combinatorial 3-designs [10; Ex. 5.2.7]. |

Self-dual linear code | Self-dual extremal codes yield combinatorial \(\leq 5\)-designs using the Assmus-Mattson theorem [8] (see [10; Sec. 5.4]). See [32; Table 1.61, pg. 683] for a table of combinatorial designs obtained from self-dual codes. |

Ternary Golay code | The supports of the weight-five (weight-six) codewords of the (extended) ternary Golay code support the Steiner system \(S(4,5,11)\) (\(S(5,6,12)\)) [15,16][17; pg. 89]. Its blocks are called hexads. |

\([2^r,2^r-r-1,4]\) Extended Hamming code | Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [17; pg. 89]. |

\([2^r-1,2^r-r-1,3]\) Hamming code | Weight-three codewords of the \([2^r-1,2^r-r-1, 3]\) Hamming code support the Steiner system \(S(2,3,2^r-1)\) [17; 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 [8]. There are several designs associated with this code [33]. |

\([7,4,3]\) 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 [10; Ex. 5.2.5]. |

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

\(q\)-ary quadratic-residue (QR) code | The supports of fixed-weight codewords of certain \(q\)-ary QR codes support combinatorial designs [8,16,27]. |

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