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Algebraic geometry code

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Algebraic geometry codes, often abbreviated AG codes, are a type of <a href="Linear_code" title="Linear code">linear code</a> that generalize <a href="Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction">Reed–Solomon codes</a>. The Russian mathematician <a href="Valery_Goppa" title="Valery Goppa">V. D. Goppa</a> constructed these codes for the first time in 1982.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

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;<a href="#cite_note-:0-2">[2]</a> however, this is no longer the standard term used in <a href="Coding_theory" title="Coding theory">coding theory</a> literature. This is due to the fact that <a href="Binary_Goppa_code" title="Binary Goppa code">Goppa codes</a> are a distinct class of codes which were also constructed by Goppa in the early 1970s.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a>

These codes attracted interest in the coding theory community because they have the ability to surpass the <a href="Gilbert%E2%80%93Varshamov_bound" title="Gilbert–Varshamov bound">Gilbert–Varshamov bound</a>; at the time this was discovered, the Gilbert–Varshamov bound had not been broken in the 30 years since its discovery.<a href="#cite_note-:1-6">[6]</a> 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".<a href="#cite_note-7">[7]</a> The name of this paper may be one source of confusion affecting references to algebraic geometry codes throughout 1980s and 1990s coding theory literature.

<div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2></div>

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.

<div class="mw-heading mw-heading3"><h3 id="Reed–Solomon_codes">Reed–Solomon codes</h3></div>

Algebraic geometry codes are a generalization of <a href="Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction">Reed–Solomon codes</a>. Constructed by <a href="Irving_S._Reed" title="Irving S. Reed">Irving Reed</a> and <a href="Gustave_Solomon" title="Gustave Solomon">Gustave Solomon</a> in 1960, Reed–Solomon codes use univariate polynomials to form codewords, by evaluating polynomials of sufficiently small degree at the points in a <a href="Finite_field" title="Finite field">finite field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

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</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy">.<a href="#cite_note-8">[8]</a>

Formally, Reed–Solomon codes are defined in the following way. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}=\{\alpha _{1},\dots ,\alpha _{q}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}=\{\alpha _{1},\dots ,\alpha _{q}\}}</annotation>

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</math><img src="./241ba31d21fce91fd7346736bef7ed20e775e9b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.028ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{q}=\{\alpha _{1},\dots ,\alpha _{q}\}}" loading="lazy">. Set positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leq n\leq q}">

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<annotation encoding="application/x-tex">{\displaystyle k\leq n\leq q}</annotation>

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</math><img src="./abde767f501eabf12c024c656025d3e2b3d6ef9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.872ex; height:2.509ex;" alt="{\displaystyle k\leq n\leq q}" loading="lazy">. Let <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}[x]_{<k}:=\left\{f\in \mathbb {F} _{q}[x]:\deg f<k\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}[x]_{<k}:=\left\{f\in \mathbb {F} _{q}[x]:\deg f<k\right\}}</annotation>

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</math>The Reed–Solomon code <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RS(q,n,k)}">

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<annotation encoding="application/x-tex">{\displaystyle RS(q,n,k)}</annotation>

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</math><img src="./0c6461c91d58eb5d332eac9cd7e899141c34aaca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.816ex; height:2.843ex;" alt="{\displaystyle RS(q,n,k)}" loading="lazy"> is the evaluation code<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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}.}">

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<annotation encoding="application/x-tex">{\displaystyle 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}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Codes_from_algebraic_curves">Codes from algebraic curves</h3></div>

Goppa observed that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

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</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> can be considered as an affine line, with corresponding <a href="Projective_line" title="Projective line">projective line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}</annotation>

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</math><img src="./6d97f98bd7959e5069a7f8f6de02c9a3b47936f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.435ex; height:3.676ex;" alt="{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}" loading="lazy">. Then, the polynomials in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}[x]_{<k}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}[x]_{<k}}</annotation>

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</math><img src="./52ce5d010b875652e08543464484d1b328905866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.399ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{q}[x]_{<k}}" loading="lazy"> (i.e. the polynomials of degree less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

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</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy">) can be thought of as polynomials with <a href="Zeros_and_poles" title="Zeros and poles">pole</a> allowance no more than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> at the <a href="Point_at_infinity" title="Point at infinity">point at infinity</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}</annotation>

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</math><img src="./6d97f98bd7959e5069a7f8f6de02c9a3b47936f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.435ex; height:3.676ex;" alt="{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}" loading="lazy">.<a href="#cite_note-:1-6">[6]</a>

With this idea in mind, Goppa looked toward the <a href="Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a>. The elements of a Riemann–Roch space are exactly those functions with pole order restricted below a given threshold,<a href="#cite_note-:2-9">[9]</a> with the restriction being encoded in the coefficients of a corresponding <a href="Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisor</a>. Evaluating those functions at the <a href="Rational_point" title="Rational point">rational points</a> on an algebraic curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

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</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> (that is, the points in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}^{2}}</annotation>

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</math><img src="./e141a3c7088ff93ff77c93aac55e55b527b33443.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.475ex; height:3.509ex;" alt="{\displaystyle \mathbb {F} _{q}^{2}}" loading="lazy"> on the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">) gives a code in the same sense as the Reed-Solomon construction.

However, because the parameters of algebraic geometry codes are connected to <a href="Algebraic_function_field" title="Algebraic function field">algebraic function fields</a>, the definitions of the codes are often given in the language of algebraic function fields over finite fields.<a href="#cite_note-:3-10">[10]</a> 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.<a href="#cite_note-:2-9">[9]</a>

Formally, algebraic geometry codes are defined in the following way.<a href="#cite_note-:3-10">[10]</a> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F/\mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle F/\mathbb {F} _{q}}</annotation>

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</math><img src="./02077ce7b233f4f1a2f0946c3acec57a7a420107.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.312ex; height:3.009ex;" alt="{\displaystyle F/\mathbb {F} _{q}}" loading="lazy"> be an algebraic function field, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=P_{1}+\dots +P_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle D=P_{1}+\dots +P_{n}}</annotation>

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</math><img src="./85627e2e8e6a30b9231e924acba26bf3eb775cdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.684ex; height:2.509ex;" alt="{\displaystyle D=P_{1}+\dots +P_{n}}" loading="lazy"> be the sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> distinct places of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F/\mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle F/\mathbb {F} _{q}}</annotation>

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</math><img src="./02077ce7b233f4f1a2f0946c3acec57a7a420107.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.312ex; height:3.009ex;" alt="{\displaystyle F/\mathbb {F} _{q}}" loading="lazy"> of degree one, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> be a divisor with disjoint <a href="Support_(mathematics)" title="Support (mathematics)">support</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">. The algebraic geometry code <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\mathcal {L}}(D,G)}">

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\mathcal {L}}(D,G):=\lbrace (f(P_{1}),\dots ,f(P_{n})):f\in {\mathcal {L}}(G)\rbrace \subseteq \mathbb {F} _{q}^{n}.}">

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<annotation encoding="application/x-tex">{\displaystyle C_{\mathcal {L}}(D,G):=\lbrace (f(P_{1}),\dots ,f(P_{n})):f\in {\mathcal {L}}(G)\rbrace \subseteq \mathbb {F} _{q}^{n}.}</annotation>

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</math>More information on these codes may be found in both introductory texts<a href="#cite_note-:1-6">[6]</a> as well as advanced texts on coding theory.<a href="#cite_note-:3-10">[10]</a><a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Reed-Solomon_codes">Reed-Solomon codes</h3></div>

One can see that

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<annotation encoding="application/x-tex">{\displaystyle RS(q,n,k)={\mathcal {C}}_{\mathcal {L}}(D,(k-1)P_{\infty })}</annotation>

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</math><img src="./d66fc7bd693c40b2d1a311c707ab5c90caf5f313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.664ex; height:2.843ex;" alt="{\displaystyle RS(q,n,k)={\mathcal {C}}_{\mathcal {L}}(D,(k-1)P_{\infty })}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> is the point at infinity on the projective line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}</annotation>

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</math><img src="./6d97f98bd7959e5069a7f8f6de02c9a3b47936f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.435ex; height:3.676ex;" alt="{\displaystyle \mathbb {P} _{\mathbb {F} _{q}}^{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=P_{1}+\dots +P_{q}}">

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<annotation encoding="application/x-tex">{\displaystyle D=P_{1}+\dots +P_{q}}</annotation>

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</math><img src="./07b3ac81ff31dafd5fe9e8f7dfb083707e8bacdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.454ex; height:2.843ex;" alt="{\displaystyle D=P_{1}+\dots +P_{q}}" loading="lazy"> is the sum of the other <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<div class="mw-heading mw-heading3"><h3 id="One-point_Hermitian_codes">One-point Hermitian codes</h3></div>

The Hermitian curve is given by the equation<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{q+1}=y^{q}+y}">

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<annotation encoding="application/x-tex">{\displaystyle x^{q+1}=y^{q}+y}</annotation>

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</math>considered over the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q^{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q^{2}}}</annotation>

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</math><img src="./34a59cf6418ca809794b79845c16b169e2426a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.247ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{q^{2}}}" loading="lazy">.<a href="#cite_note-:0-2">[2]</a> This curve is of particular importance because it meets the <a href="Hasse's_theorem_on_elliptic_curves" title="Hasse's theorem on elliptic curves">Hasse–Weil bound</a> with equality, and thus has the maximal number of affine points over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q^{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q^{2}}}</annotation>

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</math><img src="./34a59cf6418ca809794b79845c16b169e2426a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.247ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{q^{2}}}" loading="lazy">.<a href="#cite_note-12">[12]</a> With respect to algebraic geometry codes, this means that Hermitian codes are long relative to the alphabet they are defined over.<a href="#cite_note-13">[13]</a>

The Riemann–Roch space of the Hermitian function field is given in the following statement.<a href="#cite_note-:0-2">[2]</a> For the Hermitian function field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q^{2}}(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q^{2}}(x,y)}</annotation>

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</math><img src="./9a46bc624041b0395eb36090fa7bad973c683ec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.576ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{q^{2}}(x,y)}" loading="lazy"> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{q+1}=y^{q}+y}">

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<annotation encoding="application/x-tex">{\displaystyle x^{q+1}=y^{q}+y}</annotation>

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</math><img src="./af92a7b81c4d0e1aef3041a6b6bba1255726ca96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.662ex; height:3.009ex;" alt="{\displaystyle x^{q+1}=y^{q}+y}" loading="lazy"> and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {Z} ^{+}}">

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<annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} ^{+}}</annotation>

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</math><img src="./3aa93f7da8db0f7cbbaaa8ce8ef18cb50d41e129.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.942ex; height:2.509ex;" alt="{\displaystyle m\in \mathbb {Z} ^{+}}" loading="lazy">, the Riemann–Roch space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}(mP_{\infty })}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(mP_{\infty })}</annotation>

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</math><img src="./638fecebcc22ed4262ab3af129ba0fdc5c232049.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.821ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}(mP_{\infty })}" loading="lazy"> is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}(mP_{\infty })=\left\langle x^{a}y^{b}:0\leq b\leq q-1,aq+b(q+1)\leq m\right\rangle ,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(mP_{\infty })=\left\langle x^{a}y^{b}:0\leq b\leq q-1,aq+b(q+1)\leq m\right\rangle ,}</annotation>

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</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> is the point at infinity on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{q}(\mathbb {F} _{q^{2}})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{q}(\mathbb {F} _{q^{2}})}</annotation>

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</math><img src="./cbdd5c0f788084e4a2bbee5d2b71088b05eb1dd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.009ex; height:3.176ex;" alt="{\displaystyle {\mathcal {H}}_{q}(\mathbb {F} _{q^{2}})}" loading="lazy">.

With that, the one-point Hermitian code can be defined in the following way. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{q}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{q}}</annotation>

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</math><img src="./e9708dcbe8092596cd5d7afff863d9aaf28f3164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.952ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{q}}" loading="lazy"> be the Hermitian curve defined over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q^{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q^{2}}}</annotation>

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</math><img src="./34a59cf6418ca809794b79845c16b169e2426a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.247ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{q^{2}}}" loading="lazy">.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> be the point at infinity on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{q}(\mathbb {F} _{q^{2}})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{q}(\mathbb {F} _{q^{2}})}</annotation>

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</math><img src="./cbdd5c0f788084e4a2bbee5d2b71088b05eb1dd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.009ex; height:3.176ex;" alt="{\displaystyle {\mathcal {H}}_{q}(\mathbb {F} _{q^{2}})}" loading="lazy">, and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=P_{1}+\cdots +P_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle D=P_{1}+\cdots +P_{n}}</annotation>

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</math>be a divisor supported by the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:=q^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle n:=q^{3}}</annotation>

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</math><img src="./d8e4a346d003371079cfabda0657971a66efe06e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.274ex; height:3.009ex;" alt="{\displaystyle n:=q^{3}}" loading="lazy"> distinct <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q^{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q^{2}}}</annotation>

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</math><img src="./34a59cf6418ca809794b79845c16b169e2426a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.247ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{q^{2}}}" loading="lazy">-rational points on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{q}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{q}}</annotation>

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</math><img src="./e9708dcbe8092596cd5d7afff863d9aaf28f3164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.952ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}_{q}}" loading="lazy"> other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy">.

The one-point Hermitian code <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(D,mP_{\infty })}">

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<annotation encoding="application/x-tex">{\displaystyle C(D,mP_{\infty })}</annotation>

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</math><img src="./ecc76f9727f44cbe7f178cb4e46e572b5173bf6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.942ex; height:2.843ex;" alt="{\displaystyle C(D,mP_{\infty })}" loading="lazy"> is

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<annotation encoding="application/x-tex">{\displaystyle 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}.}</annotation>

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</math><img src="./de81ebd313bf1fcb44bc684c2dda3a4bffeab082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:58.303ex; height:3.509ex;" alt="{\displaystyle 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}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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