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{{Short description|Mathematical linear code}}
'''Algebraic geometry codes''', often abbreviated AG codes, are a type of [[linear code]] that generalize [[Reed–Solomon error correction|Reed–Solomon codes]]. The Russian mathematician [[Valery Goppa|V. D. Goppa]] constructed these codes for the first time in 1982.<ref>{{Cite journal |last=Goppa |first=Valerii Denisovich |author-link=Valery Goppa |date=1982 |title=Algebraico-geometric codes |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1646&option_lang=eng |journal=Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya |volume=46 |issue=4 |pages=726–781 |via=Russian Academy of Sciences, Steklov Mathematical Institute of Russian}}</ref>

== 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;<ref name=":0">{{Cite journal |last=Stichtenoth |first=Henning |date=1988 |title=A note on Hermitian codes over GF(q^2) |url=https://ieeexplore.ieee.org/abstract/document/21267 |journal=IEEE Transactions on Information Theory |volume=34 |issue=5 |pages=1345–1348 |via=IEEE}}</ref> however, this is no longer the standard term used in [[coding theory]] literature. This is due to the fact that [[Binary Goppa code|Goppa codes]] are a distinct class of codes which were also constructed by Goppa in the early 1970s.<ref>{{Cite journal |last=Goppa |first=Valery Denisovich |author-link=Valery Goppa |date=1970 |title=A new class of linear error-correcting codes |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=1748&option_lang=eng |journal=Probl. Inf. Transm. |volume=6 |pages=300–304}}</ref><ref>{{Cite journal |last=Goppa |first=Valerii Denisovich |author-link=Valery Goppa |date=1972 |title=Codes Constructed on the Base of (L,g)-Codes |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=791&option_lang=eng |journal=Problemy Peredachi Informatsii |volume=8 |issue=2 |pages=107–109 |via=Russian Academy of Sciences, Branch of Informatics, Computer Equipment and}}</ref><ref>{{Cite journal |last=Berlekamp |first=Elwyn |author-link=Elwyn Berlekamp |date=1973 |title=Goppa codes |url=https://ieeexplore.ieee.org/abstract/document/1055088 |journal=IEEE Transactions on Information Theory |volume=19 |issue=5 |pages=590–592 |via=IEEE}}</ref>

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.<ref name=":1">{{Cite book |last=Walker |first=Judy L. |title=Codes and Curves |publisher=American Mathematical Society |year=2000 |isbn=0-8218-2628-X |location= |pages=15}}</ref> 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".<ref>{{Cite journal |last=Tsfasman |first=Michael |last2=Vladut |first2=Serge |last3=Zink |first3=Thomas |author-link3=Thomas Zink |date=1982 |title=Modular curves, Shimura curves, and Goppa codes better than the Varshamov-Gilbert bound |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19821090103 |journal=Mathematische Nachrichten}}</ref> 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 error correction|Reed–Solomon codes]]. Constructed by [[Irving S. Reed|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]] <math>\mathbb{F}_q</math>.<ref>{{Cite journal |last=Reed |first=Irving |author-link=Irving S. Reed |last2=Solomon |first2=Gustave |author-link2=Gustave Solomon |date=1960 |title=Polynomial codes over certain finite fields |url=https://epubs.siam.org/doi/abs/10.1137/0108018?journalCode=smjmap.1 |journal=Journal of the Society for Industrial and Applied Mathematics |volume=8 |issue=2 |pages=300–304 |via=SIAM}}</ref>

Formally, Reed–Solomon codes are defined in the following way. Let <math>\mathbb{F}_q=\{\alpha_1, \dots, \alpha_q\}</math>. Set positive integers <math>k \leq n \leq q</math>. Let <math display="block">\mathbb{F}_{q}[x]_{<k} := \left\{ f \in \mathbb{F}_q[x]: \deg f < k \right\}</math>The Reed–Solomon code <math>RS(q,n,k)</math> is the evaluation code<math display="block">RS(q,n,k) = \left\{ \left( f(\alpha_1), f(\alpha_2), \dots, f(\alpha_n) \right) : f \in \mathbb{F}_{q}[x]_{<k} \right\} \subseteq \mathbb{F}_q^{n}.</math>

=== Codes from algebraic curves ===
Goppa observed that <math>\mathbb{F}_q</math> can be considered as an affine line, with corresponding [[projective line]] <math>\mathbb{P}^1_{\mathbb{F}_q}</math>. Then, the polynomials in <math>\mathbb{F}_{q}[x]_{<k}</math> (i.e. the polynomials of degree less than <math>k</math> over <math>\mathbb{F}_q</math>) can be thought of as polynomials with [[Zeros and poles|pole]] allowance no more than <math>k</math> at the [[point at infinity]] in <math>\mathbb{P}^1_{\mathbb{F}_q}</math>.<ref name=":1" />

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,<ref name=":2">{{Cite journal |last=Hoholdt |first=Tom |last2=van Lint |first2=Jacobus |author-link2=J. H. van Lint |last3=Pellikaan |first3=Ruud |date=1998 |title=Algebraic geometry codes |url=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 |journal=Handbook of coding theory |volume=1 |issue=Part 1 |pages=871–961 |via=Elsevier Amsterdam}}</ref> with the restriction being encoded in the coefficients of a corresponding [[Divisor (algebraic geometry)|divisor]]. Evaluating those functions at the [[rational point]]s on an algebraic curve <math>X</math> over <math>\mathbb{F}_q</math> (that is, the points in <math>\mathbb{F}_q^2</math> on the curve <math>X</math>) 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 field]]s, the definitions of the codes are often given in the language of algebraic function fields over finite fields.<ref name=":3">{{Cite book |last=Stichtenoth |first=Henning |title=Algebraic function fields and codes |publisher=Springer Science & Business Media |year=2009 |isbn=978-3-540-76878-4 |edition=2nd |pages=45–65}}</ref> 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.<ref name=":2" />

Formally, algebraic geometry codes are defined in the following way.<ref name=":3" /> Let <math>F / \mathbb{F}_q</math> be an algebraic function field, <math>D = P_1 + \dots + P_n</math> be the sum of <math>n</math> distinct places of <math>F / \mathbb{F}_q</math> of degree one, and <math>G</math> be a divisor with disjoint [[Support (mathematics)|support]] from <math>D</math>. The algebraic geometry code <math>C_{\mathcal{L}}(D,G)</math> associated with divisors <math>D</math> and <math>G</math> is defined as <math display="block">C_{\mathcal{L}}(D,G) := \lbrace (f(P_1),\dots,f(P_n)) : f \in \mathcal{L}(G) \rbrace \subseteq \mathbb{F}_q^n.</math>More information on these codes may be found in both introductory texts<ref name=":1" /> as well as advanced texts on coding theory.<ref name=":3" /><ref>{{Cite book |last=van Lint |first=Jacobus |title=Introduction to coding theory |publisher=Springer |year=1999 |isbn=978-3-642-63653-0 |edition=3rd |pages=148–166}}</ref>

== Examples ==

=== Reed-Solomon codes ===
One can see that

<math>RS(q,n,k) = \mathcal{C}_\mathcal{L}(D,(k-1)P_\infty)</math>

where <math>P_\infty</math> is the point at infinity on the projective line <math>\mathbb{P}^1_{\mathbb{F}_q}</math> and <math>D = P_1 + \dots + P_q</math> is the sum of the other <math>\mathbb{F}_q</math>-rational points.

=== One-point Hermitian codes ===
The Hermitian curve is given by the equation<math display="block">x^{q+1} = y^q + y</math>considered over the field <math>\mathbb{F}_{q^2}</math>.<ref name=":0" /> This curve is of particular importance because it meets the [[Hasse's theorem on elliptic curves|Hasse–Weil bound]] with equality, and thus has the maximal number of affine points over <math>\mathbb{F}_{q^2}</math>.<ref>{{Cite journal |last=Garcia |first=Arnoldo |author-link=Arnaldo Garcia |last2=Viana |first2=Paulo |date=1986 |title=Weierstrass points on certain non-classical curves |url=https://link.springer.com/article/10.1007/BF01200462 |journal=Archiv der Mathematik |volume=46 |pages=315–322 |via=Springer}}</ref> With respect to algebraic geometry codes, this means that Hermitian codes are long relative to the alphabet they are defined over.<ref>{{Cite journal |last=Tiersma |first=H.J. |date=1987 |title=Remarks on codes from Hermitian curves |url=https://ieeexplore.ieee.org/abstract/document/1057327 |journal=IEEE Transactions on Information Theory |volume=33 |issue=4 |pages=605–609 |via=IEEE}}</ref>

The Riemann–Roch space of the Hermitian function field is given in the following statement.<ref name=":0" /> For the Hermitian function field <math>\mathbb{F}_{q^2}(x,y)</math> given by <math>x^{q+1} = y^q + y</math> and for <math>m \in \mathbb{Z}^+</math>, the Riemann–Roch space <math>\mathcal{L}(mP_\infty)</math> is<math display="block">\mathcal{L}(mP_\infty) = \left\langle x^a y^b : 0 \leq b \leq q-1, aq + b(q+1) \leq m \right\rangle ,</math>where <math>P_\infty</math> is the point at infinity on <math>\mathcal{H}_q(\mathbb{F}_{q^2})</math>.

With that, the one-point Hermitian code can be defined in the following way. Let <math>\mathcal{H}_q</math> be the Hermitian curve defined over <math>\mathbb{F}_{q^2}</math>.

Let <math>P_\infty</math> be the point at infinity on <math>\mathcal{H}_q(\mathbb{F}_{q^2})</math>, and <math display="block">D = P_1 + \cdots + P_n</math>be a divisor supported by the <math>n := q^3</math> distinct <math>\mathbb{F}_{q^2}</math>-rational points on <math>\mathcal{H}_q</math> other than <math>P_\infty</math>.

The one-point Hermitian code <math>C(D,mP_\infty)</math> is

<math>C(D,mP_\infty) := \left\lbrace (f(P_1),\dots,f(P_n)) : f \in \mathcal{L}(mP_\infty) \right\rbrace \subseteq \mathbb{F}_{q^2}^n.</math>

== References ==
{{Reflist}}

[[Category:Coding theory]]
[[Category:Algebraic curves]]
[[Category:Finite fields]]
[[Category:Articles containing proofs]]

Algebraic geometry code - Revision history

imported>Kku: link coding theory