Here is a list of projective codes.

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

Cycle LDPC code | An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns. |

Cycle code | A code whose parity-check matrix forms the incidence matrix of a graph. This code's properties are derived from the size two chain complex associated with the graph. |

Denniston code | Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{GF(2^a)}\) codes for \(i < a\) constructed using Denniston arcs. |

Glynn code | The unique trace-Hermitian self-dual \([10,5,6]_9\) code, constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve. |

Hexacode | The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [1], and conformal field theory [2]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [3]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\). |

Hill projective-cap code | Member of a projective code family that contains of \(q\)-ary sharp configurations and that is constructed using projective caps. |

Hoffman-Singleton cycle code | A \([50,21,12]\) cycle code whose parity-check matrix is the incidence matrix of the Hoffman-Singleton graph [4]. Its dual is a \([50,29,8]\) code [5; Table II]. |

Hyperoval code | A projective code constructed using hyperovals in projective space. |

Incidence-matrix projective code | Code whose generator matrix is the incidence matrix of points and hyperplanes in a projective space. Has been generalized to incidence matrices of other structures [6,7][8; Sec. 14.4]. Columns of a code's parity-check matrix can similarly correspond to an incidence matrix. |

Margulis LDPC code | Member of a class of LDPC codes deterministically constructed using a class of regular graphs with no short cycles. Related explicit LDPC constructions [9] utilize Ramanujan graphs [10,11]. |

Ovoid code | Member of a \([q^2+1,4,q^2-q]_q\) projective code family that is universally optimal and that is constructed using ovoids in projective space. See [12; pg. 107][13; pg. 192] for further details. |

Petersen cycle code | A \([15,6,5]\) cycle code whose parity-check matrix is the incidence matrix of the Petersen graph. The Petersen graph can be thought of as a dodecahedron with antipodes identified [14; Appx. A.2.1]. |

Projective geometry code | Linear \(q\)-ary \([n,k,d]\) code such that columns of its generator matrix \(G\) does not contain any repeated columns or the zero column. That way, each column corresponds to a distinct point in the projective space \(PG(k-1,q)\) arising from a \(k\)-dimensional vector space over \(GF(q)\). If the columns are linearly independent, then the codewords are collectively called an information set. Columns of a code's parity-check matrix can similarly correspond to points in projective space. This formulation yields connections to projective geometry, which can be applied to determine code properties. |

Projective two-weight code | A projective code whose codewords all have one of two possible nonzero Hamming weights. |

Tetracode | The \([4,2,3]_3\) self-dual MDS code that has connections to lattices [1]. |

\([2^m-1,m,2^{m-1}]\) simplex code | A member of the code family that is dual to the \([2^m,2^m-m-1,3]\) Hamming family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping. |

\([56,6,36]_3\) Hill-cap code | Projective two-weight ternary code based on the Games graph [16][15; Table 19.1] and Hill's 56-cap [17]. Its automorphism group contains \(PSL_3(4)\) [18]. |

\([7,3,4]\) simplex code | Second-smallest member of the simplex code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping. |

\([78,6,56]_4\) Hill-cap code | Projective two-weight quaternary code based on a cap that corresponds to a strongly regular graph [16; Table 7.1]. |

\(q\)-ary simplex code | An \([n,m,q^{m-1}]_q\) projective code with \(n=\frac{q^m-1}{q-1}\), denoted as \(S(q,m)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,q)\), with each column being a chosen representative of the corresponding element. |

## References

- [1]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [2]
- J. A. Harvey and G. W. Moore, “Moonshine, superconformal symmetry, and quantum error correction”, Journal of High Energy Physics 2020, (2020) arXiv:2003.13700 DOI
- [3]
- G. Hoehn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
- [4]
- A. J. Hoffman and R. R. Singleton, “On Moore Graphs with Diameters 2 and 3”, IBM Journal of Research and Development 4, 497 (1960) DOI
- [5]
- V. D. Tonchev, “Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs”, IEEE Transactions on Information Theory 43, 1021 (1997) DOI
- [6]
- B. Bagchi and S. P. Inamdar, “Projective Geometric Codes”, Journal of Combinatorial Theory, Series A 99, 128 (2002) DOI
- [7]
- M. Lavrauw, L. Storme, and G. Van de Voorde (2010). Linear codes from projective spaces. In A. Bruen & D. Wehlau (Eds.), Contemporary Mathematics (Vol. 523, pp. 185–202). Providence, RI, USA: American Mathematical Society (AMS).
- [8]
- L. Storme, "Coding Theory and Galois Geometries." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [9]
- J. Rosenthal and P. O. Vontobel, “Constructions of regular and irregular LDPC codes using Ramanujan graphs and ideas from Margulis”, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252) DOI
- [10]
- A. Lubotzky, R. Phillips, and P. Sarnak, “Ramanujan graphs”, Combinatorica 8, 261 (1988) DOI
- [11]
- G. Davidoff, P. Sarnak, and A. Valette, Elementary Number Theory, Group Theory and Ramanujan Graphs (Cambridge University Press, 2001) DOI
- [12]
- R. Calderbank and W. M. Kantor, “The Geometry of Two-Weight Codes”, Bulletin of the London Mathematical Society 18, 97 (1986) DOI
- [13]
- J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
- [14]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [15]
- A. E. Brouwer, "Two-weight Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [16]
- R. A. Games, “The packing problem for projective geometries over GF(3) with dimension greater than five”, Journal of Combinatorial Theory, Series A 35, 126 (1983) DOI
- [17]
- Hill, R. (1973). On the largest size of cap in s53. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, 54(3), 378-384.
- [18]
- N. Pace and A. Sonnino, “On linear codes admitting large automorphism groups”, Designs, Codes and Cryptography 83, 115 (2016) DOI