Here is a list of algebraic-geometry codes constructed using algebraic curves.
| Code | Description |
|---|---|
| Algebraic-geometry (AG) code | Binary or \(q\)-ary code or subcode constructed from an algebraic curve of some genus over a finite field via the evaluation construction, the residue construction, or more general constructions that yield nonlinear codes. Linear AG codes from the first two constructions are also called geometric Goppa codes. |
| Cartier code | A generalization of the Goppa codes to codes defined from curves of non-zero genus. Each code is a subcode of a certain residue AG code and is constructed using the Cartier operator. |
| 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. |
| 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. |
| 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. |
| 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 \( R_a(x) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i} \). |
| 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 \(q^3 + 1\). |
| 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 \(GF(4)\). |
| 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 \(x^3 y + y^3 z + z^3 x = 0\) ([4], Ex. 2.75). |
| Narrow-sense RS code | An \([q-1,k,n-k+1]_q\) RS code whose points \(\alpha_i\) are all \((i-1)\)st powers of a primitive element \(\alpha\) of \(GF(q)\). |
| Nonlinear AG code | Nonlinear \(q\)-ary code constructed by evaluating functions on an algebraic curve. |
| Norm-trace code | Evaluation AG code of rational functions evaluated on points lying on a Miura-Kamiya curve in either affine or projective space. The family is named as such because the equations defining the curves can be expressed in terms of the field norm and field trace. |
| 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. |
| Primitive narrow-sense BCH code | BCH codes for \(b=1\) and for \(n=q^r-1\) for some \(r\geq 2\). |
| Reed-Solomon (RS) code | An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\). |
| 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 \(\omega\). |
| 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 \(S_n\). |
| Srivastava code | A special case of a generalized Srivastava code for \(z_j = \alpha_j^{\mu}\) for some \(\mu\) and \(t=1\). |
| 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. 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 [1]. |
| 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. |
| \([2^r-1,2^r-r-1,3]\) Hamming code | Member of an infinite family of perfect linear codes with parameters \([2^r-1,2^r-r-1, 3]\) for \(r \geq 2\). Their \(r \times (2^r-1) \) parity-check matrix \(H\) has all possible non-zero \(r\)-bit strings as its columns. Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code. |
| \([7,4,3]\) Hamming code | Second-smallest member of the Hamming code family. |
| \(q\)-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]
- 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]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.