Here is a list of codes related to self-dual classical codes.

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

Combinatorial design code | A constant-weight binary code that is mapped into a combinatorial \(t\)-design. |

Cyclic linear \(q\)-ary code | A \(q\)-ary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=q^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes. |

Divisible code | A linear \(q\)-ary block code is \(\Delta\)-divisible if the Hamming weight of each of its codewords is divisible by divisor \(\Delta\). A \(2\)-divisible (\(4\)-divisible, \(8\)-divisible) code is called even (doubly-even, triply-even) [1,2]. A code is called singly-even if all codewords are even and at least one has weight equal to 2 modulo 4. More generally, a code is \(m\)-even if it is \(2^{m}\)-divisible. |

Dodecacode | The unique trace-Hermitian self-dual additive \((12,4^6,6)_4\) code. Its codewords are cyclic permutations of \((\omega 10100100101)\), where \(GF(4)=\{0,1,\omega,\bar{\omega}\}\) [3; Sec. 2.4.8]. Another generator matrix can be found in [4; Ex. 9.10.8]. |

Dual linear code | For any \([n,k]_q\) linear code \(C\), the dual (or orthogonal) code, \begin{align} C^\perp = \{ y\in GF(q)^{n} ~|~ x\cdot y=0 \forall x\in C\}, \tag*{(1)}\end{align} where the ordinary, standard, or Euclidean inner product is \(x\cdot y = \sum_{i=1}^n x_i y_i\) for coordinates \(x_i,y_i\). |

Golay code | A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [2] and sporadic simple groups [5]. 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 [6]. Shortening the Golay code yields the \([22,10,8]\), \([22,11,7]\), and \([22,12,6]\) shortened Golay codes [7]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [8,9]. |

Hexacode | The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [2], and conformal field theory [10]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [11]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\). |

Jump code | A CE code designed to detect and correct amplitude damping errors. An \(((n,K))\) jump code is denoted as \(((n,K,t))_w\) (which conflicts with modular-qudit notation), where \(t\) is the maximum number of qubits that can be corrected after each one has undergone a jump error \(|0\rangle\langle 1|\), and where each codeword is a uniform superposition of qubit basis states with Hamming weight \(w\). |

Niemeier lattice code | One of the 24 positive-definite even unimodular lattices of rank 24. |

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. |

Octacode | The unique self-dual linear code of length 8 and Lee distance 6 over \(\mathbb{Z}_4\) with generator matrix \begin{align} \begin{pmatrix} 3 & 3 & 2 & 3 & 1 & 0 & 0 & 0\\ 3 & 0 & 3 & 2 & 3 & 1 & 0 & 0\\ 3 & 0 & 0 & 3 & 2 & 3 & 1 & 0\\ 3 & 0 & 0 & 0 & 3 & 2 & 3 & 1 \end{pmatrix}\,. \tag*{(2)}\end{align} |

Pless symmetry code | A member of a family of self-dual ternary \([2q+2,q+1]_3\) codes for any power of an odd prime satisfying \(q \equiv 2\) modulo 3. |

Self-dual additive code | An additive \((n,2^n)_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product, usually the trace-Hermitian inner product. |

Self-dual code over \(R\) | An additive linear code \(C\) over a ring \(R\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product. |

Self-dual linear code | An \([n,n/2]_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to an inner product, most commonly either Euclidean or Hermitian. Self-dual codes exist only for even lengths and have dimension \(k=n/2\). A code that is equivalent to its dual via permutations is called iso-dual. |

Ternary Golay code | A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [2] and sporadic simple groups [5]. Adding a parity bit to the code results in the self-dual \([12,6,6]_3\) extended ternary Golay code. Up to equivalence, both codes are unique for their respective parameters [6]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode. |

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

Unimodular lattice code | A lattice that is equal to its dual, \(L^\perp = L\). Unimodular lattices have \(\det L = \pm 1\). |

\([48,24,12]\) self-dual code | An extended quadratic-residue code that is known to be the only self-dual doubly-even code at its parameters [12]. |

## References

- [1]
- S. Kurz, “Divisible Codes”, (2023) arXiv:2112.11763
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- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
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- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
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- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
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- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
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- P. Delsarte and J. M. Goethals, “Unrestricted codes with the golay parameters are unique”, Discrete Mathematics 12, 211 (1975) DOI
- [7]
- J. H. Conway and N. J. A. Sloane, “Orbit and coset analysis of the Golay and related codes”, IEEE Transactions on Information Theory 36, 1038 (1990) DOI
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- W. Feit. Some remarks on weight functions of spaces over GF(2), unpublished (1972)
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- C. L. Mallows and N. J. A. Sloane, “Weight enumerators of self-orthogonal codes”, Discrete Mathematics 9, 391 (1974) DOI
- [10]
- 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
- [11]
- G. Hoehn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
- [12]
- S. K. Houghten et al., “The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code”, IEEE Transactions on Information Theory 49, 53 (2003) DOI