Algebraic geometry code

Short description: Mathematical linear code

Algebraic geometry codes, often abbreviated AG codes, are a type of linear code that generalize Reed–Solomon codes. The Russian mathematician V. D. Goppa constructed these codes for the first time in 1982.[1]

## Contents

* 1 History
* 2 Construction
* 2.1 Reed–Solomon codes
* 2.2 Codes from algebraic curves
* 3 Decoding
* 4 Examples
* 4.1 Reed-Solomon codes
* 4.2 One-point Hermitian codes
* 5 References

## History

The name of these codes has evolved since the publication of Goppa's paper describing them. Historically these codes have also been referred to as geometric Goppa codes;[2] however, this is no longer the standard term used in coding theory literature. This is due to the fact that Goppa codes are a distinct class of codes which were also constructed by Goppa in the early 1970s.[3][4][5]

These codes attracted interest in the coding theory community because they have the ability to surpass the Gilbert–Varshamov bound; at the time this was discovered, the Gilbert–Varshamov bound had not been broken in the 30 years since its discovery.[6] This was demonstrated by Tfasman, Vladut, and Zink in the same year as the code construction was published, in their paper "Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound".[7] The name of this paper may be one source of confusion affecting references to algebraic geometry codes throughout 1980s and 1990s coding theory literature.

## Construction

In this section the construction of algebraic geometry codes is described. The section starts with the ideas behind Reed–Solomon codes, which are used to motivate the construction of algebraic geometry codes.

### Reed–Solomon codes

Algebraic geometry codes are a generalization of Reed–Solomon codes. Constructed by Irving Reed and Gustave Solomon in 1960, Reed–Solomon codes use univariate polynomials to form codewords, by evaluating polynomials of sufficiently small degree at the points in a finite field 𝔽q.[8]

Formally, Reed–Solomon codes are defined in the following way. Let 𝔽q={α1,…,αq}. Set positive integers k≤n≤q. Let 𝔽q[x]<k:={f∈𝔽q[x]:deg⁡f<k}The Reed–Solomon code RS(q,n,k) is the evaluation codeRS(q,n,k)={(f(α1),f(α2),…,f(αn)):f∈𝔽q[x]<k}⊆𝔽qn.

### Codes from algebraic curves

Goppa observed that 𝔽q can be considered as an affine line, with corresponding projective line ℙ𝔽q1. Then, the polynomials in 𝔽q[x]<k (i.e. the polynomials of degree less than k over 𝔽q) can be thought of as polynomials with pole allowance no more than k at the point at infinity in ℙ𝔽q1.[6]

With this idea in mind, Goppa looked toward the Riemann–Roch theorem. The elements of a Riemann–Roch space are exactly those functions with pole order restricted below a given threshold,[9] with the restriction being encoded in the coefficients of a corresponding divisor. Evaluating those functions at the rational points on an algebraic curve X over 𝔽q (that is, the points in 𝔽q2 on the curve X) gives a code in the same sense as the Reed-Solomon construction.

However, because the parameters of algebraic geometry codes are connected to algebraic function fields, the definitions of the codes are often given in the language of algebraic function fields over finite fields.[10] Nevertheless, it is important to remember the connection to algebraic curves, as this provides a more geometrically intuitive method of thinking about AG codes as extensions of Reed-Solomon codes.[9]

Formally, algebraic geometry codes are defined in the following way.[10] Let F/𝔽q be an algebraic function field, D=P1+…+Pn be the sum of n distinct places of F/𝔽q of degree one, and G be a divisor with disjoint support from D. The algebraic geometry code Cℒ(D,G) associated with divisors D and G is defined as Cℒ(D,G):={(f(P1),…,f(Pn)):f∈ℒ(G)}⊆𝔽qn.More information on these codes may be found in both introductory texts[6] as well as advanced texts on coding theory.[10][11]

## Decoding

Some algebraic geometry codes can be decoded using algorithms based on the Berlekamp–Massey–Sakata algorithm. Shojiro Sakata introduced this algorithm as an extension of the Berlekamp–Massey algorithm to multidimensional arrays.[12] In particular, one-point algebraic geometry codes can be decoded efficiently using the Berlekamp–Massey–Sakata algorithm, and later variants have been developed for multipoint codes from algebraic curves.[13]

## Examples

### Reed-Solomon codes

One can see that

RS(q,n,k)=𝒞ℒ(D,(k−1)P∞)

where P∞ is the point at infinity on the projective line ℙ𝔽q1 and D=P1+…+Pq is the sum of the other 𝔽q-rational points.

### One-point Hermitian codes

The Hermitian curve is given by the equationxq+1=yq+yconsidered over the field 𝔽q2.[2] This curve is of particular importance because it meets the Hasse–Weil bound with equality, and thus has the maximal number of affine points over 𝔽q2.[14] With respect to algebraic geometry codes, this means that Hermitian codes are long relative to the alphabet they are defined over.[15]

The Riemann–Roch space of the Hermitian function field is given in the following statement.[2] For the Hermitian function field 𝔽q2(x,y) given by xq+1=yq+y and for m∈ℤ+, the Riemann–Roch space ℒ(mP∞) isℒ(mP∞)=⟨xayb:0≤b≤q−1,aq+b(q+1)≤m⟩,where P∞ is the point at infinity on ℋq(𝔽q2).

With that, the one-point Hermitian code can be defined in the following way. Let ℋq be the Hermitian curve defined over 𝔽q2.

Let P∞ be the point at infinity on ℋq(𝔽q2), and D=P1+⋯+Pnbe a divisor supported by the n:=q3 distinct 𝔽q2-rational points on ℋq other than P∞.

The one-point Hermitian code C(D,mP∞) is

C(D,mP∞):={(f(P1),…,f(Pn)):f∈ℒ(mP∞)}⊆𝔽q2n.

## References

1. ↑ Goppa, Valerii Denisovich (1982). "Algebraico-geometric codes". Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 46 (4): 726–781. https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1646&option_lang=eng.
2. ↑ 2.0 2.1 2.2 Stichtenoth, Henning (1988). "A note on Hermitian codes over GF(q^2)". IEEE Transactions on Information Theory 34 (5): 1345–1348. https://ieeexplore.ieee.org/abstract/document/21267.
3. ↑ Goppa, Valery Denisovich (1970). "A new class of linear error-correcting codes". Probl. Inf. Transm. 6: 300–304. https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=1748&option_lang=eng.
4. ↑ Goppa, Valerii Denisovich (1972). "Codes Constructed on the Base of (L,g)-Codes". Problemy Peredachi Informatsii 8 (2): 107–109. https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=791&option_lang=eng.
5. ↑ Berlekamp, Elwyn (1973). "Goppa codes". IEEE Transactions on Information Theory 19 (5): 590–592. https://ieeexplore.ieee.org/abstract/document/1055088.
6. ↑ 6.0 6.1 6.2 Walker, Judy L. (2000). Codes and Curves. American Mathematical Society. pp. 15. ISBN 0-8218-2628-X.
7. ↑ Tsfasman, Michael; Vladut, Serge; Zink, Thomas (1982). "Modular curves, Shimura curves, and Goppa codes better than the Varshamov-Gilbert bound". Mathematische Nachrichten. https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19821090103.
8. ↑ Reed, Irving; Solomon, Gustave (1960). "Polynomial codes over certain finite fields". Journal of the Society for Industrial and Applied Mathematics 8 (2): 300–304. https://epubs.siam.org/doi/abs/10.1137/0108018?journalCode=smjmap.1.
9. ↑ 9.0 9.1 Hoholdt, Tom; van Lint, Jacobus; Pellikaan, Ruud (1998). "Algebraic geometry codes". Handbook of coding theory 1 (Part 1): 871–961. https://www.researchgate.net/profile/R-Pellikaan/publication/293123965_Algebraic_geometry_of_codes_handbook_of_coding_theory/links/56c59cc708ae7fd4625baa21/Algebraic-geometry-of-codes-handbook-of-coding-theory.pdf.
10. ↑ 10.0 10.1 10.2 Stichtenoth, Henning (2009). Algebraic function fields and codes (2nd ed.). Springer Science & Business Media. pp. 45–65. ISBN 978-3-540-76878-4.
11. ↑ van Lint, Jacobus (1999). Introduction to coding theory (3rd ed.). Springer. pp. 148–166. ISBN 978-3-642-63653-0.
12. ↑ Sakata, Shojiro (February 1990). "Extension of the Berlekamp-Massey algorithm to N dimensions". Information and Computation 84 (2): 207–239. doi:10.1016/0890-5401(90)90039-K.
13. ↑ Sakata, Shojiro; Fujisawa, Masaya (April 2014). "Fast decoding of multipoint codes from algebraic curves". IEEE Transactions on Information Theory 60 (4): 2054–2064. doi:10.1109/TIT.2014.2300473.
14. ↑ Garcia, Arnoldo; Viana, Paulo (1986). "Weierstrass points on certain non-classical curves". Archiv der Mathematik 46: 315–322. https://link.springer.com/article/10.1007/BF01200462.
15. ↑ Tiersma, H.J. (1987). "Remarks on codes from Hermitian curves". IEEE Transactions on Information Theory 33 (4): 605–609. https://ieeexplore.ieee.org/abstract/document/1057327.

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