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Algebraic geometry code

Algebraic geometry codes, often abbreviated as AG codes, are a class of linear error-correcting codes defined over finite fields \mathbb{F}_q by evaluating sections of line bundles (or rational functions) on the rational points of an , generalizing classical constructions like Reed-Solomon and Goppa codes to achieve superior asymptotic performance. These codes were first conceptualized by V.D. Goppa in his 1981 paper "Codes on s," where he proposed using the function field of an to construct codes with parameters determined by s and evaluation maps at rational points. In the construction, for a smooth projective curve X of g over \mathbb{F}_q, a D = \sum_{i=1}^n P_i supported on n distinct rational points P_i, and another G with support disjoint from D, the code C_\mathcal{L}(D, G) is the image of the evaluation map \text{ev}: \mathcal{L}(G) \to \mathbb{F}_q^n, where \mathcal{L}(G) is the Riemann-Roch space of G. The dual code C_\Omega(D, G) is similarly defined using differentials. The dimension k of C_\mathcal{L}(D, G) satisfies k = \ell(G) - \ell(G - D) by the Riemann-Roch theorem, often approximating \deg(G) - g + 1 for large \deg(G), while the minimum distance d obeys the designed distance bound d \geq n - \deg(G). These parameters satisfy the k + d \leq n + 1, but AG codes excel asymptotically: in 1982, M.A. Tsfasman, S.G. Vlăduţ, and Th. Zink constructed families of codes attaining the Tsfasman-Vlăduţ-Zink (TVZ) bound, stating that for q a and relative distance \delta = d/n, the asymptotic rate R = k/n satisfies R \geq 1 - \delta - \frac{1}{\sqrt{q-1}}, surpassing the Gilbert-Varshamov bound for q \geq 49. Notable examples include Hermitian codes from the curve x^{q+1} + y^{q+1} + z^{q+1} = 0 over \mathbb{F}_{q^2}, which provide explicit constructions with efficient decoding algorithms up to roughly half the designed using methods like list decoding or the Feng-Rao bound. AG codes have influenced applications in , storage systems, and , with ongoing research extending them to higher-dimensional varieties and generalized constructions.

Introduction and Fundamentals

Definition and basic concepts

Algebraic geometry codes, often abbreviated as AG codes, are linear error-correcting codes over a \mathbb{F}_q constructed as evaluation codes using points and functions on algebraic varieties, particularly nonsingular projective curves, thereby generalizing classical codes like Reed–Solomon codes. These codes leverage the structure of the function field of the variety to achieve parameters that surpass certain bounds for random linear codes, especially for large lengths. Consider a nonsingular projective X of g defined over \mathbb{F}_q, with rational points P_1, \dots, P_n on X. Let D = P_1 + \cdots + P_n be an effective of n, representing the evaluation points, and let G be another of m with support disjoint from that of D. The Riemann–Roch space L(G) is the \mathbb{F}_q- of all rational functions f in the function field of X (including the zero function) such that the principal (f) satisfies (f) + G \geq 0, meaning the poles of f are controlled by G. By the , the dimension k = \ell(G) = \dim_{\mathbb{F}_q} L(G) equals m - g + 1 when m \geq 2g - 1. The AG code C_L(D, G) is the image of the \mathbb{F}_q-linear evaluation map \mathrm{ev}_D \colon L(G) \to \mathbb{F}_q^n, \quad f \mapsto (f(P_1), \dots, f(P_n)), so the codewords are precisely these evaluations of basis functions from L(G) at the points in the support of D. Thus, C_L(D, G) forms a linear [n, k, d]_q code, where the generator matrix has rows given by the evaluations of a basis of L(G) at P_1, \dots, P_n. The minimum distance d satisfies the designed distance bound d \geq n - m = n - \deg(G), derived from the fact that any nonzero f \in L(G) vanishes at at most \deg(G) points. Reed–Solomon codes correspond to the special case where X is the projective line (of genus g = 0) over \mathbb{F}_q. For higher-dimensional varieties, the construction extends analogously by evaluating sections of line bundles (or sheaves) at rational points, using higher-codimension divisors and appropriate cohomology spaces in place of L(G).

Mathematical prerequisites

Algebraic geometry codes build upon foundational concepts from coding theory. A linear code over the finite field \mathbb{F}_q is defined as a k-dimensional subspace C of the vector space \mathbb{F}_q^n. Such codes are characterized by the parameters [n, k, d], where n denotes the length (dimension of the ambient space), k the dimension of the code (number of information symbols), and d the minimum between any two distinct codewords. The generator matrix G is a k \times n matrix whose rows form a basis for C, while the parity-check matrix H is an (n-k) \times n matrix satisfying cH^T = 0 for all codewords c \in C. A fundamental limitation is given by the , which asserts that d \leq n - k + 1 for any . Key elements from algebraic geometry are also essential. Projective varieties over \mathbb{F}_q are the common zero loci of homogeneous polynomials in projective space \mathbb{P}^m(\mathbb{F}_q). Algebraic curves are irreducible projective varieties of dimension one, each associated with a genus g \geq 0, an invariant that quantifies the curve's topological complexity; for example, the projective line has genus 0, while elliptic curves have genus 1. The function field \mathbb{F}_q(X) of a curve X consists of all rational functions on X, which are quotients of polynomials regular away from finitely many poles. Divisors on X are formal \mathbb{Z}-linear combinations \sum_{P \in X} n_P P of points P on the curve, with the degree of a divisor D defined as \deg(D) = \sum n_P. The Riemann-Roch theorem relates the geometry of the to the dimensions of spaces of functions with controlled poles. For a D on a of g with \deg(D) \geq 2g - 2, \dim L(D) = \deg(D) - g + 1 + i(D), where L(D) is the of rational functions f such that \operatorname{div}(f) + D \geq 0 (the Riemann-Roch space) and i(D) is the index of specialty. The theorem holds in a more general form as l(D) - l(K - D) = \deg(D) - g + 1, where K is a canonical divisor, l(E) = \dim L(E) for any divisor E, and the difference l(D) - l(K - D) captures the adjustment from the naive degree-genus relation. This result will later help bound the dimensions of spaces used in code constructions. A crucial quantity is the number of \mathbb{F}_q-rational points on a curve X of genus g, denoted \#X(\mathbb{F}_q), which counts the solutions to the defining equations over \mathbb{F}_q. The Weil bound provides an estimate: \left| \#X(\mathbb{F}_q) - (q + 1) \right| \leq 2g \sqrt{q}. This inequality limits how many points a curve can have over \mathbb{F}_q, influencing the potential length of associated codes.

Historical Development

Origins and Goppa codes

The origins of algebraic geometry codes trace back to the work of V. D. Goppa in the early 1980s, building on his earlier invention of classical Goppa codes. In 1970, Goppa introduced classical Goppa codes as a class of linear error-correcting codes constructed using polynomials over finite fields, specifically leveraging the rational function field \mathbb{F}_q(t) where places correspond to the zeros and poles of rational functions. These codes generalized earlier constructions like Reed–Solomon codes by employing more flexible polynomial structures to achieve good error-correcting capabilities. Seeking to improve code parameters beyond the limitations of the rational function field, Goppa extended this framework in his 1981 paper by incorporating algebraic curves over finite fields, using Riemann surfaces and function fields of general curves to define a broader class of codes now known as geometric Goppa codes. This generalization allowed for the evaluation of functions at more rational points on the curve compared to the single-variable case, motivated by the goal of constructing codes that surpass the Gilbert-Varshamov bound through the geometric properties of curves with higher . Unlike classical Goppa codes, which are confined to the places of \mathbb{F}_q(t) and thus limited in the number of evaluation points, geometric Goppa codes exploit the richer structure of arbitrary algebraic curves, enabling potentially superior asymptotic performance. In a follow-up 1982 paper, Goppa specifically demonstrated the construction of such codes using hyperelliptic curves, further illustrating how the divisor-based approach on these curves yields linear codes with parameters tied to the curve's geometry. Over time, the terminology evolved from "geometric Goppa codes," emphasizing their connection to Goppa's classical work, to the more general "," reflecting their reliance on broader tools from beyond just curves.

Key breakthroughs and advancements

A landmark advancement in the theory of algebraic geometry codes occurred in with the work of Tsfasman, Vlăduţ, and Zink, who constructed sequences of codes from modular curves that exceed the asymptotic Gilbert-Varshamov bound, attaining the TVZ bound R \geq 1 - \delta - \frac{1}{\sqrt{q-1}} for relative distance \delta = d/n < 1 - \frac{1}{\sqrt{q-1}} - \epsilon for any \epsilon > 0, where R = k/n is the rate, marking the first time codes outperformed the GV bound using geometric constructions for square q \geq 49. This result highlighted the potential of algebraic curves to yield asymptotically good codes, particularly for square field sizes q = p^2. The TVZ construction's impact extended to the study of asymptotic optimality, where Ihara's constant A(q) = \limsup_{g \to \infty} N_q(X)/g(X)—the supremum ratio of rational points N_q(X) to g(X) over curves X—plays a central role. For fixed q, as block length n \to \infty, algebraic geometry codes achieve rates and distances approaching the Drinfeld-Vlăduţ upper bound on A(q) \leq 1 - 1/\sqrt{q}, with TVZ providing explicit families attaining near-optimal values and demonstrating asymptotic optimality in this regime. In the , further progress focused on explicit constructions, notably using Drinfeld modular curves as analogues of classical modular towers for general finite fields, enabling polynomial-time computable codes with parameters matching or approaching TVZ bounds without relying on square q. These developments, building on Drinfeld modules, facilitated practical implementations and extended the applicability of algebraic geometry codes to diverse field characteristics. The saw extensions to higher-dimensional varieties, such as surfaces and abelian varieties, where codes from ideals in coordinate rings of higher-genus objects offered improved trade-offs in dimension and minimum distance, though with increased . Concurrently, list-decoding algorithms for algebraic geometry codes emerged, adapting interpolation-based methods to decode up to a constant fraction of errors beyond half the minimum distance, significantly enhancing error-correcting capabilities over unique decoding. By the 2020s, algebraic geometry codes integrated with , providing asymptotically good stabilizer codes for fault-tolerant , including applications in with optimal scaling and decoded quantum protocols that leverage their geometric structure for robust phase estimation.

Construction Principles

General framework for codes

The general framework for evaluation codes establishes a paradigm for constructing linear codes over a \mathbb{F}_q by evaluating functions from a at a finite set of points on an algebraic variety. Consider a smooth projective variety X defined over \mathbb{F}_q, a finite set of distinct \mathbb{F}_q-rational points Z = \{P_1, \dots, P_n\} \subset X(\mathbb{F}_q), and a finite-dimensional \mathbb{F}_q- V of \mathbb{F}_q-rational functions on X. The evaluation map \ev_Z: V \to \mathbb{F}_q^n is defined by \ev_Z(f) = (f(P_1), \dots, f(P_n)) for f \in V, and the resulting code C is the image of this map, a linear subspace of \mathbb{F}_q^n with length n. This construction captures a broad class of codes, including generalizations of Reed-Solomon codes where V consists of polynomials of bounded degree evaluated on field elements viewed as points on the projective line. In algebraic geometry codes, this framework is formalized using s to specify both the evaluation points and the . Let D = \sum_{i=1}^n P_i be the effective supported on Z, so n = \deg D. Choose a rational G on X such that \supp(G) \cap Z = \emptyset, and set V = \mathcal{L}(G) = \{ f \in k(X) \mid \div(f) + G \geq 0 \} \cup \{ 0 \}, the Riemann-Roch space of global sections of the associated to G, where k(X) is the function field of X. The is then C(D, G) = \ev_Z(\mathcal{L}(G)), with dimension k = \dim_{\mathbb{F}_q} \mathcal{L}(G). By the Riemann-Roch theorem, for varieties where it applies (such as curves), k \geq \deg G + 1 - g, with g denoting the or an analogous irregularity measure. The parameters of such codes follow from geometric properties: the length is n = \deg D, the dimension is approximately k \approx \deg G - g + 1, and the minimum distance satisfies d \geq n - \deg G, known as the designed distance \delta = n - \deg G. This distance bound arises because any nonzero f \in \mathcal{L}(G) has at most \deg G zeros (counting multiplicity), ensuring that \ev_Z(f) has weight at least n - \deg G. A refinement of the , accounting for the geometry, gives d \geq n - k + 1 - g, reflecting the overhead due to the variety's complexity compared to one-dimensional cases like Reed-Solomon codes. These parameters position algebraic geometry codes as powerful constructions exceeding classical bounds in certain regimes.

Codes from algebraic curves

Algebraic geometry codes from algebraic curves are constructed using a nonsingular projective algebraic curve X of genus g defined over a finite field \mathbb{F}_q. A set of n distinct rational points P_1, \dots, P_n \in X(\mathbb{F}_q) is selected, and the divisor D = P_1 + \dots + P_n is formed. A divisor G on X is chosen such that its support is disjoint from the support of D, ensuring that the evaluation map is well-defined. The code C_L(D, G) is then defined as the image of the evaluation map \mathrm{ev}_D: L(G) \to \mathbb{F}_q^n, where L(G) = \{ f \in \mathbb{F}_q(X) \mid \mathrm{div}(f) + G \geq 0 \} \cup \{0\} is the Riemann-Roch space associated to G, consisting of rational functions on X with poles controlled by G. This construction generalizes classical evaluation codes by incorporating the geometry of the curve, allowing for improved parameters through the interplay of points and function spaces. The dimension k and minimum distance d of the code C_L(D, G) are determined via the Riemann-Roch theorem applied to the curve. Specifically, the dimension satisfies k = \ell(G) - \ell(G - D), where \ell(\cdot) denotes the dimension of the corresponding Riemann-Roch space; if \deg(G) < n, then \ell(G - D) = 0, yielding k = \ell(G). For \deg(G) \geq 2g - 1, the Riemann-Roch theorem gives \ell(G) = \deg(G) - g + 1. The minimum distance follows a designed bound analogous to the BCH bound, d \geq n - \deg(G), which arises because any function in L(G) can vanish at most \deg(G) points unless identically zero. These parameters enable codes that surpass certain classical bounds for sufficiently large n relative to q. A common specialization is the one-point code, where G = m P_\infty for some rational point P_\infty not among the P_i and integer m > 2g - 2. Here, \ell(G) = m - g + 1, so the code has dimension k = m - g + 1 and designed distance d \geq n - m. This setup simplifies the choice of G while leveraging the full strength of for large m, and it is particularly effective on curves with many rational points. An illustrative example is provided by the Hermitian curve, defined over \mathbb{F}_{q^2} by the equation x^{q+1} + y^q + y = 0, which has genus g = q(q-1)/2. This curve admits exactly q^3 + 1 rational points over \mathbb{F}_{q^2}; for one-point Hermitian codes, n = q^3 points are typically selected, excluding the point at infinity P_\infty = (0:1:0), with D their sum and G = m P_\infty. The resulting codes achieve dimension k = m - g + 1 and distance d \geq q^3 - m for m \geq 2g - 1, demonstrating asymptotic optimality in certain regimes. For hyperelliptic curves, given by an affine model y^2 = f(x) of degree $2g+1 or $2g+2 over \mathbb{F}_q (with projective closure), an explicit basis for L(G) can be constructed using the structure of the function field. For G = m P_\infty where P_\infty is the hyperelliptic branch point at infinity, the basis consists of monomials x^i for i = 0, \dots, \lfloor m/2 \rfloor and x^i y for i = 0, \dots, \lfloor (m-1)/2 \rfloor, respecting the pole orders determined by the valuation at P_\infty; this extends to differentials for dual code constructions, where \Omega(G - D) uses holomorphic differentials adjusted by G - D. Such bases facilitate efficient encoding and decoding for these codes.

Codes from higher-dimensional varieties

Algebraic geometry codes from higher-dimensional varieties extend the evaluation code framework to projective varieties X of dimension r \geq 2 defined over the \mathbb{F}_q. The construction involves selecting a set S \subseteq X(\mathbb{F}_q) of rational points on X and a \mathcal{O}_X(D) associated to an effective D. The code is then the image of the evaluation map ev_S: H^0(X, \mathcal{O}_X(D)) \to \mathbb{F}_q^n, where n = |S|, yielding a over \mathbb{F}_q of length n. A primary challenge in these constructions arises from the scarcity of \mathbb{F}_q-rational points on higher-dimensional varieties compared to curves, with the Lang-Weil estimate bounding the number of points by approximately q^r + O(q^{r-1/2}). To address this, methods such as adjunction—intersecting X with curves to apply one-dimensional techniques—or the use of complete linear systems for divisors are employed; additional approaches include deriving codes from Grassmannians, which parameterize subspaces and yield MDS codes with parameters [ \binom{m}{\ell}_q, \ell(m - \ell) + 1, q^{\ell(m - \ell)} ], and from toric varieties, linking to lattice-based toric codes. The dimension k of the code equals \dim H^0(X, \mathcal{O}_X(D)), computed via the Hirzebruch-Riemann-Roch theorem, which expresses it in terms of the Chern character of \mathcal{O}_X(D) and the Todd class of X: \chi(\mathcal{O}_X(D)) = \int_X \operatorname{ch}(\mathcal{O}_X(D)) \cdot \operatorname{td}(T_X), where \chi approximates k under vanishing higher . Distance bounds are generally weaker than those for curve codes due to the more complex geometry, analogous to higher-genus effects; volume-based estimates from the refine this to forms like d \geq n - c \cdot \deg(D)^{r/(r+1)} for suitable constants c. In the , seminal constructions on Hermitian surfaces provided concrete examples with improved performance. Consider the surface X \subset \mathbb{P}^3 defined over \mathbb{F}_{r^2} by the equation x_0^{r+1} + x_1^{r+1} + x_2^{r+1} + x_3^{r+1} = 0; evaluating sections of \mathcal{O}_X(1) at all rational points yields a code with parameters [n = (r^2 + 1)(r^3 + 1), k = 4, d = r^5 + r^2], achieving relative rates superior to those from algebraic curves for certain square q = r^2.

Key Properties

Code parameters and bounds

Algebraic geometry (AG) codes are evaluation codes defined on the rational points of an , typically a of g > 0 over a \mathbb{F}_q. The length n of an AG code C_L(D, G) is the number of \mathbb{F}_q-rational points in the support of the divisor D, so \deg D = n. The k is given by k = \ell(G) - \ell(G - D), where \ell denotes the dimension of the Riemann-Roch space; when \deg G < n, this simplifies to k = \ell(G). By the Riemann-Roch theorem, \ell(G) = \deg G - g + 1 + \ell(K - G) for a canonical K, yielding k \approx \deg G - g + 1 when \deg G \geq 2g - 1 and \ell(K - G) = 0. The designed minimum distance \delta for C_L(D, G) is \delta = n - \deg G, reflecting the maximum number of zeros a non-zero function in L(G) can have at points in the support of D. The actual minimum distance d satisfies d \geq n - \deg G, a bound derived from the fact that any non-constant f \in L(G) can vanish at most \deg G points in the support of D, as the degree of its zero divisor equals the degree of its pole divisor (bounded by \deg G). Tighter values of d can be obtained from weight enumerators, which count codewords of each weight, or from advanced algebraic geometry bounds such as the order bound or Feng-Rao bound. A fundamental bound is the geometric Singleton bound, d \geq n - k + 1 - g, which refines the classical d \geq n - k + 1 by incorporating the genus g. This improvement arises from applying the Riemann-Roch theorem to the effective divisors associated with codewords and is particularly effective for high-genus curves, where g > 0 allows AG codes to outperform Reed-Solomon codes in asymptotic rate-distance tradeoffs. For instance, when the Riemann-Roch space achieves its generic dimension, the geometric Singleton bound aligns with the designed bound d \geq n - \deg G. Proofs of the minimum often rely on geometric invariants like the gonality \gamma of the curve—the minimal of a rational map to \mathbb{P}^1—or adjoint conditions tied to the . The gonality provides lower bounds via the d = \min \{ n - \deg E \mid E \text{ effective}, \ell(G - E) > 0 \}, where low-gonality curves limit the degrees of such E. Adjoint conditions for the dual code C_\Omega(D, G) involve the failure of linear equivalence to a divisor plus an effective divisor disjoint from points, yielding d^* \geq \deg G - 2g + 2 for the designed distance of the dual. A related bound is d \geq n - \delta + 1 - g, adjusting the designed distance for geometric effects. For one-point AG codes, where D = (n-1)P_\infty for a rational point P_\infty and G = m P_\infty, explicit formulas for k and d stem from the Weierstrass gap theorem at P_\infty. The theorem states there are exactly g gaps—integers r with \ell(r P_\infty) = \ell((r-1) P_\infty)—and the dimension is k = m + 1 - \#\{\text{gaps } \leq m\}, precisely counting basis functions with poles up to order m at P_\infty. The minimum distance follows from analyzing the zero multiplicities at affine rational points, often yielding d = n - \nu_m, where \nu_m is the largest non-gap at or below m, enabling exact computation for specific curves like elliptic or hyperelliptic ones.

Asymptotic performance

In the asymptotic regime of (AG) codes, one considers families of codes where the block length n tends to infinity. The relative is defined as \delta = d/n, where d is the minimum , and the is R = k/n, where k is the dimension. The optimal asymptotic performance is captured by the function A_q(\delta) = \sup \{ R : \text{there exists a family of } [n, k, d]_q \text{ codes with } k/n \to R, d/n \to \delta \}, representing the maximum achievable for a given relative over the \mathbb{F}_q. The Tsfasman-Vlăduţ-Zink (TVZ) bound provides a fundamental lower bound on A_q(\delta). Specifically, if the Ihara constant A(q) > 1, then A_q(\delta) \geq 1 - \delta - 1/A(q) for $0 \leq \delta \leq 1 - 1/A(q). For q \geq 49 a perfect square, this bound exceeds the Gilbert-Varshamov bound A_q(\delta) \geq 1 - H_q(\delta), where H_q(x) = x \log_q (q-1) - x \log_q x - (1-x) \log_q (1-x) is the q-ary function, particularly for \delta < 1 - 1/\sqrt{q}. This superiority holds because the TVZ construction yields families of codes with rates strictly greater than $1 - H_q(\delta) - \epsilon for any \epsilon > 0 in that range. The Ihara constant A(q) = \limsup_{g \to \infty} \frac{N_q(g)}{g}, where N_q(g) is the maximum number of \mathbb{F}_q-rational points on an of g, directly influences AG code performance. For families of curves yielding good AG codes, the number of evaluation points n scales asymptotically with the g as n \approx A(q) g, while the designed distance relates to g, enabling rates and distances tied to A(q). The Drinfeld-Vladut bound states that A(q) \leq \sqrt{q} - 1 for all q. Modular tower constructions achieve the optimal A(q) = \sqrt{q} - 1 when q is a perfect square, as shown using Drinfeld modular curves, which realize the TVZ bound explicitly. These towers provide sequences of function fields with the required point-genus growth, leading to AG code families that are asymptotically optimal in the specified regime. Advancements from 2017 have extended explicit tower constructions to block-transitive AG codes, attaining the TVZ bound over \mathbb{F}_{q^2}. As of 2025, AG codes have been applied to decoded quantum , enhancing fault-tolerant protocols.

Decoding Methods

Algorithms up to half the minimum

Standard decoding algorithms for (AG) codes correct up to t = \lfloor (d-1)/2 \rfloor errors, where d is the minimum , relying on computations and solutions to equations derived from the of the underlying . These methods extend the Patterson algorithm originally developed for Goppa codes, which computes s S_i = \sum y_j \alpha_j^i for received vector y and distinct points \alpha_j, then solves for the error locator via square-root computations in the field modulo the Goppa . This approach was generalized to AG codes from arbitrary curves by Ehrhard, who formulated a equation solvable through linear algebra over Riemann-Roch spaces, enabling error location and evaluation without explicit square roots. For AG codes defined on curves, the decoding process begins by computing syndromes from the received word y, typically as residues or evaluations in the space of differentials or adjoints associated with the code's divisor. The core step involves solving a generalized key equation relating the syndrome to the error evaluator and locator within the Riemann-Roch space L((2t-2)P_\infty), where P_\infty is the point at infinity. Once solved, the roots of the error locator identify error positions via interpolation on the curve, and the error evaluator yields error values, achieving unique decoding up to the designed distance. The Reed-Solomon code serves as a special case where the curve degenerates to the projective line, reducing to the Berlekamp-Massey algorithm. The of these syndrome-based decoders is generally O(n^2) operations in the base field, where n is the code length, arising from calculation and solves over spaces of roughly $2t; improvements to O(n \log^2 n) are possible using fast multiplication and Fourier-like transforms adapted to the curve's or residue maps. For Hermitian codes over \mathbb{F}_{q^2}, defined on the curve x^{q+1} + y^{q+1} + z^{q+1} = 0, explicit algorithms leverage the curve's symmetry to compute q-ary s directly from evaluations at rational points, solving the key equation via identities or Gröbner bases for locators in time up to half the .

Advanced and list decoding techniques

Advanced decoding techniques for algebraic geometry (AG) codes extend beyond the unique decoding radius of half the minimum distance, enabling correction of a larger fraction of errors at the cost of potentially outputting a list of candidate codewords rather than a unique one. The seminal Guruswami-Sudan algorithm generalizes the Sudan interpolation-based list decoding from Reed-Solomon codes to AG codes defined on algebraic curves of arbitrary genus, achieving a list-decoding radius up to the Johnson bound while producing a polynomial-sized list in the block length n. In this framework, decoding involves finding a nonzero bivariate polynomial Q(x, y) over the function field, with degree at most (1 - R)n in x and \sqrt{kn} in y (where R = k/n is the design rate and k the dimension), such that Q(P_i, y_i) = 0 for all received symbols y_i at evaluation points P_i, followed by factorization to recover possible messages. For Hermitian codes, a prominent class of AG codes on curves over \mathbb{F}_{q^2}, this approach corrects up to n - \sqrt{k n} errors, approaching the theoretical limit for list decoding. For one-point AG codes, the Feng-Rao decoding algorithm improves upon standard methods by exploiting the structure of the dual code to correct errors up to the designed minimum distance d = n - k + 1 - g (where g is the genus), which exceeds half the distance for low-rate codes, achieving up to n - \sqrt{nk} errors in certain regimes through majority-logic decoding of unknown syndromes. An algebraic-geometric adaptation of the Koetter-Vardy soft-decision decoding algorithm further enhances performance by incorporating reliability information from the channel, constructing a tree of interpolation polynomials on the curve to list-decode beyond hard-decision limits, with complexity polynomial in n. Post-2010 advancements include adaptations of decoding algorithms to codes on curves, particularly Hermitian codes, where low-complexity variants generate test vectors from the most unreliable positions and apply list decoding to each, yielding significant gains over Koetter-Vardy with reduced computational overhead. In 2025, decoded quantum emerged as a novel application, leveraging the of codes—such as Hermitian codes—for error correction in quantum optimization tasks, mapping decoding problems to quantum transform-based protocols that reduce requirements by one-third compared to Reed-Solomon-based approaches.

Examples

Reed–Solomon codes

Reed–Solomon codes provide a foundational example of algebraic geometry codes, specifically evaluation codes derived from the over a \mathbb{F}_q. In this construction, the underlying curve is the \mathbb{P}^1_{\mathbb{F}_q}, which is a smooth projective curve of genus 0. The code is defined by selecting n = q-1 distinct evaluation points \alpha_1, \dots, \alpha_n \in \mathbb{F}_q \setminus \{0\}, corresponding to the divisor D = \sum_{i=1}^n P_i where P_i is the point at \alpha_i. The code space is then the Riemann–Roch space L(G) for the divisor G = (k-1) \infty, where \infty denotes the point at on \mathbb{P}^1, yielding functions that are polynomials of degree less than k. The codewords are obtained via evaluation: for f \in L(G), the codeword is \mathrm{ev}(f) = (f(\alpha_1), \dots, f(\alpha_n)) \in \mathbb{F}_q^n. These codes achieve length n = q-1, dimension k, and minimum distance d = n - k + 1, saturating the Singleton bound for maximum distance separable (MDS) codes. The generator matrix G for the Reed–Solomon code has entries G_{i,j} = \alpha_j^{i-1} for i = 1, \dots, k and j = 1, \dots, n, reflecting the monomial basis of polynomials of degree less than k. This structure ensures optimal error-correcting capability up to \lfloor (n-k)/2 \rfloor errors, making them ideal for applications requiring high reliability over finite fields. An extension of Reed–Solomon codes incorporates the point at , increasing the length to n = q while preserving the MDS property, with evaluation now including a suitable value at \infty (often defined via ). This extended version maintains the parameters [q, k, q - k + 1] and is particularly useful in scenarios demanding full field utilization.

Hermitian codes

Hermitian codes are a family of algebraic geometry codes constructed from the Hermitian curve, providing a non-trivial example beyond Reed–Solomon codes due to the positive of the underlying curve. The Hermitian curve is defined by the affine equation y^q + y = x^{q+1} over the \mathbb{F}_{q^2}, where q is a . This curve has g = \frac{q(q-1)}{2} and exactly q^3 affine \mathbb{F}_{q^2}-rational points, which serve as the evaluation points for the code. The point at infinity P_\infty is the unique hyperflex at infinity, with pole orders v_{P_\infty}(x) = -q and v_{P_\infty}(y) = -(q+1). The one-point Hermitian code is the evaluation code at the affine rational points, with evaluation divisor consisting of these q^3 points and pole divisor G = m P_\infty for a positive integer m. The codewords are formed by evaluating a basis of the Riemann–Roch space L(G), which is spanned by monomials x^i y^j satisfying i q + j (q+1) \leq m. By the , the dimension is k = m - g + 1 when m \geq 2g - 2. The designed minimum distance satisfies d \geq n - \deg G = q^3 - m, reflecting the bound from the of the , though the true distance can be larger in certain ranges. The rate of the code is approximately R \approx m / q^3. This family achieves more rational points than curves of genus zero while maintaining good relative distance, making it suitable for applications requiring longer codes over \mathbb{F}_{q^2}.

Other notable codes

Hyperelliptic codes are constructed from hyperelliptic curves defined by the equation y^2 + h(x) y = f(x), where h(x) and f(x) are polynomials over a \mathbb{F}_q with q even, and the degree of f(x) is $2g + 1 for genus g = (\deg f - 1)/2. These codes are particularly suitable for even fields and offer parameters comparable to those from Hermitian curves, while providing explicit bases for the Riemann-Roch spaces that facilitate computational efficiency. Codes from modular curves, such as towers of X_0(N) for increasing N, yield asymptotically good constructions that achieve the Tsfasman-Vlăduţ-Zink (TVZ) bound on the tradeoff between code rate and relative minimum distance. These towers over finite fields of size q_0^2 produce codes with length n \approx q^{1 + 1/(2 \lfloor \log_q N \rfloor)}, enabling high rates for large genus while maintaining strong error-correcting capabilities. The Garcia-Stichtenoth codes arise from explicit towers of function fields over finite fields of square cardinality q = r^2 with r > 2, constructed recursively to attain the optimal asymptotic bound A(q) = 1/(\sqrt{q} - 1), where A(q) denotes the limit of the maximal code rate as genus tends to infinity. This tower provides a concrete family of s yielding algebraic geometry codes that surpass the Gilbert-Varshamov bound for sufficiently large parameters. Two-point codes on the Hermitian , using divisors of the form aP + bQ where P and Q are distinct rational points, achieve minimum distances strictly larger than those of the corresponding one-point codes for a range of designed distances, thereby improving error-correcting performance without increasing redundancy.

Applications

Error correction and storage

Algebraic geometry (AG) codes have found practical applications in classical error correction for communications systems, particularly where high-rate, long-length codes are required. Their asymptotic goodness allows them to outperform BCH and Reed-Solomon (RS) codes for extended code lengths, enabling more efficient data transmission over noisy channels. For instance, AG codes based on algebraic curves achieve better relative minimum distances as the field size q grows, making them suitable for scenarios demanding robust performance without exponentially increasing the alphabet size. In deep-space probes and links, AG codes support high-rate error correction by leveraging their superior parameters for reliable transmission of data across vast distances with low signal-to-noise ratios. These codes provide enhanced error-detecting and correcting capabilities compared to traditional , contributing to the integrity of mission-critical communications. evaluations have demonstrated that certain AG codes surpass codes in coding gain, such as achieving approximately 1/3 dB better performance than RS(511,127) for specific parameters. For data storage systems, AG codes enhance reliability in and configurations, where they effectively mitigate burst errors—clusters of consecutive errors arising from physical wear or interference—outperforming in certain operational regimes due to their structured algebraic properties. This burst-error resilience stems from the geometric construction, allowing targeted decoding without excessive overhead. In optical storage applications, Hermitian codes, a prominent class of AG codes derived from the Hermitian over \mathbb{F}_{q^2}, correct up to 10-15% errors for code lengths n \approx 10^6, offering scalable protection for high-density media. Comparatively, AG codes outperform random linear codes even at finite lengths when q is large, as their minimum distance exceeds the through explicit curve-based constructions, providing deterministic guarantees absent in probabilistic random ensembles. This advantage is particularly evident in storage contexts requiring predictable error thresholds.

Cryptography and secret sharing

Algebraic geometry (AG) codes have been adapted to variants of the , where the of an AG code, such as a Hermitian code, serves as the basis for , with security relying on the hardness of the . In this setup, the public key consists of a scrambled G_{\text{pub}} = S G P, where G is the of the AG code, S is an invertible scrambling matrix, and P is a , while appends an error vector of weight up to half the minimum distance to a codeword. The for general linear codes is NP-complete, providing the foundational security assumption for these systems. Constructions of such AG-based McEliece variants emerged in the , leveraging the efficient decoding algorithms available for AG codes up to half their minimum distance, with security grounded in the of decoding linear codes and related problems like finding minimum-weight codewords. Hermitian codes, as a prominent class of AG codes defined over the Hermitian curve x^{q+1} + y^{q+1} + z^{q+1} = 0 in over \mathbb{F}_{q^2}, have been particularly used due to their good parameters, such as length n = q^3 and dimension scaling favorably with the . Post-2009 developments include quantum-resistant variants employing structured subfield subcodes of Hermitian AG codes to enhance efficiency while preserving security against known attacks on the decoding problem. In secret sharing protocols, AG codes enable ramp schemes that support complex access structures, allowing reconstruction of a secret from t shares out of n while providing privacy against fewer than t shares, with information rates exceeding $1/2. These schemes distribute shares s_i = f(P_i) for distinct points P_i on an algebraic curve and a random function f drawn from the Riemann-Roch space L(G) associated with a divisor G, such that the secret can be reconstructed via interpolation when sufficiently many points are available. s_i = f(P_i), \quad f \in L(G) Reconstruction is possible if the number of shares meets or exceeds the degree threshold implicit in the divisor, enabling high-rate ramp schemes over constant-sized fields using constructions like Garcia-Stichtenoth towers. List decoding techniques can enhance robustness in these schemes by allowing recovery even with some corrupted shares.

Recent developments and emerging uses

In recent advancements, (AG) codes have been applied to , particularly in enabling fault-tolerant measurements through decoded quantum . A 2025 study demonstrates the use of Hermitian codes—a class of AG codes—for this purpose, achieving efficient decoding that supports quantum speedups in tasks over Hermitian curves. This approach leverages list recovery techniques on the duals of Hermitian codes to solve the Hermitian Optimal problem, outperforming classical algorithms like Prange's method in large parameter regimes. AG codes have also seen integration into network coding for distributed storage systems, where they facilitate reduced repair bandwidth through locality properties. Constructions based on elliptic curves, a subclass of AG codes, yield maximum distance separable (MDS) codes suitable for erasure coding in such systems, approaching optimal lengths of (q + 1 + ⌊2√q⌋)/2 while supporting efficient node repairs. These codes enhance reliability in large-scale storage, though direct deployments vary. In applications, AG codes inspire secure frameworks, particularly for tasks central to . A 2025 IEEE paper proposes AG codes for secure distributed matrix multiplication, ensuring privacy and efficiency in computationally intensive algorithms used in and model training. Additionally, 2020s research incorporates AG codes with to bolster data privacy in , enabling cross-subspace alignment for without revealing objectives. To address needs, explicit constructions of rank-metric codes from Drinfeld modules have emerged as high-impact contributions since 2020. A NSF award funds research on these codes, deriving new primitives for code-based encryption resistant to quantum attacks by analogizing methods over function fields.