Here is a list of evaluation codes.

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

Classical Goppa code | Let \( G(x) \) be a polynomial describing a projective-plane curve with coefficients from \( GF(q^m) \) for some fixed integer \(m\). Let \( L \) be a finite subset of the extension field \( GF(q^m) \) where \(q\) is prime, meaning \( L = \{\alpha_1, \cdots, \alpha_n\} \) is a subset of nonzero elements of \( GF(q^m) \). A Goppa code \( \Gamma(L,G) \) is an \([n,k,d]_q\) linear code consisting of all vectors \(a = a_1, \cdots, a_n\) such that \( R_a(x) =0 \) modulo \(G(x)\), where \( R_a(x) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i} \). |

Complete-intersection RM-type code | Evaluation code of polynomials evaluated on points lying on a complete intersection. |

Deligne-Lusztig code | Evaluation code of polynomials evaluated on points lying on a Deligne-Lusztig curve. |

Elliptic code | Evaluation AG code of rational functions evaluated on points lying on an elliptic curve, i.e., a curve of genus one. |

Evaluation AG code | Evaluation code over \(GF(q)\) on a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) whose corresponding vector space \(L\) of functions \(f\) consists of certain polynomials or rational functions. |

Evaluation code | Code whose codewords are evaluations of functions at certain fixed points. Code properties can be inferred from the structure of the functions and the underlying geometric object containing the points, often using results from algebraic geometry. |

Extended GRS code | A GRS code with an additional parity-check coordinate with corresponding evaluation point of zero. In other words, an \([n+1,k,n-k+2]_q\) GRS code whose polynomials are evaluated at the points \((\alpha_1,\cdots,\alpha_n,0)\). The case when \(n=q-1\), multipliers \(v_i=1\), and \(\alpha_i\) are \(i-1\)st powers of a primitive \(n\)th root of unity is an extended narrow-sense RS code. |

Flag-variety code | Evaluation code of polynomials evaluated on points lying on a flag variety. |

Generalized RM (GRM) code | Reed-Muller code GRM\(_q(r,m)\) of length \(n=q^m\) over \(GF(q)\) with \(0\leq r\leq m(q-1)\). Its codewords are evaluations of the set of all degree-\(\leq r\) polynomials in \(m\) variables at the points of \(GF(q)\). |

Generalized RS (GRS) code | An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors. |

Grassmannian code | Evaluation code of polynomials evaluated on points lying on a Grassmannian \({\mathbb{G}}(\ell,m)\) [1]. |

Hansen toric code | Evaluation code of a linear space of polynomials evaluated on points lying on an affine or projective toric variety. If the space is taken to be all polynomials up to some degree, the code is called a toric RM-type code of that degree. |

Hermitian code | Evaluation AG code of rational functions evaluated on points lying on a Hermitian curve in either affine or projective space. Hermitian codes improve over RS codes in length: that RS codes have length at most \(q+1\) while Hermitian codes have length \(q^3 + 1\). |

Hermitian-hypersurface code | Evaluation code of polynomials evaluated on points lying on a Hermitian hypersurface. |

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

Klein-quartic code | Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying on the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) ([5], Ex. 2.75). |

Plane-curve code | Evaluation AG code of bivariate polynomials of some finite maximum degree, evaluated at points lying on an affine or projective plane curve. |

Polynomial evaluation code | Evaluation code of polynomials (or, more generally, rational functions) at points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) on an algebraic variety \(\cal X\) of dimension greater than one (i.e., not an algebraic curve). |

Projective RM (PRM) code | Reed-Muller code for nonzero points \(\{\alpha_1,\cdots,\alpha_n\}\) with \(n=m+1\) whose leftmost nonzero coordinate is one, corresponding to an evaluation code of polynomials over projective coordinates. |

Quadric code | Evaluation code of polynomials evaluated on points lying on a quadric hypersurface. |

Reed-Muller (RM) code | Member of the RM\((r,m)\) family of linear binary codes derived from multivariate polynomials. The code parameters are \([2^m,\sum_{j=0}^{r} {m \choose j},2^{m-r}]\), where \(r\) is the order of the code satisfying \(0\leq r\leq m\). First-order RM codes are also called biorthogonal codes, while \(m\)th order RM codes are also called universe codes. Punctured RM codes RM\(^*(r,m)\) are obtained from RM codes by deleting one coordinate from each codeword. |

Reed-Solomon (RS) code | An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\). |

Repetition code | \([n,1,n]\) binary linear code encoding one bit of information into an \(n\)-bit string. The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information. The idea is to increase the code distance by repeating the logical information several times. It is a \((n,1)\)-Hamming code. Its automorphism group is \(S_n\). |

Residue AG code | Linear \(q\)-ary code defined using a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) and a linear space \(\Omega\) of certain rational differential forms \(\omega\). |

Ruled-surface code | Evaluation code of polynomials evaluated on points lying on a ruled surface. |

Schubert code | Evaluation code of polynomials evaluated on points lying on a Schubert variety. |

Segre-variety RM-type code | Evaluation code of polynomials evaluated on points lying on a Segre variety. |

Single parity-check (SPC) code | An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit or parity 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. Its codewords are all even-weight binary strings. Its automorphism group is \(S_n\). |

Suzuki-curve code | Evaluation AG code of rational functions evaluated on points lying on a Suzuki curve. |

Tamo-Barg code | A family of \(q\)-ary polynomial evaluation codes that are optimal LRCs and for which \(q\) is comparable to \(n\). |

Tamo-Barg-Vladut code | Polynomial evaluation code on algebraic curves, such as Hermitian or Garcia-Stichtenoth curves, that is constructed to be an LRC code. Codes can be constructed to be be able to recover locally after one or more erasures as well as to tackle the availability problem. |

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

Tsfasman-Vladut-Zink (TVZ) code | Member of a family of residue AG or, more generally, evaluation AG codes where \(\cal X\) is either Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve. |

\([2^m,m+1,2^{m-1}]\) First-order RM code | A member of the family of first-order RM codes. Its codewords are the rows of the \(2^m\)-dimensional Hadamard matrix \(H\) and its negation \(-H\) with the mapping \(+1\to 0\) and \(-1\to 1\). They form a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping. |

\([8,4,4]\) 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 parity-check 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

- [1]
- D. Yu. Nogin, “Codes associated to Grassmannians”, Arithmetic, Geometry, and Coding Theory DOI
- [2]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [3]
- 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
- [4]
- G. Hoehn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
- [5]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.