Algebraic Geometry Codes over Finite Fields

Algebraic geometry codes, commonly referred to as AG codes, form a class of linear error-correcting codes constructed from algebraic curves defined over finite fields. By selecting a divisor on a non-singular projective curve and evaluating functions from an associated Riemann–Roch space at a prescribed set of rational points, one obtains a code whose length, dimension and minimum distance are determined by the geometry of the curve and the choice of divisor. The central appeal of AG codes lies in their ability to surpass classical bounds for block codes, such as the Gilbert–Varshamov bound, by exploiting curves with many rational points relative to their genus. Seminal work demonstrated that families of codes derived from modular curves and Drinfeld modular towers achieve asymptotically optimal parameters, approaching the Tsfasman–Vladut–Zink limit. In practice, AG codes have found applications in deep-space communication, data storage systems and post-quantum cryptography. Recent advances have focused on multi-point constructions, explicit characterisations of Weierstrass semigroups, and novel decoding algorithms that leverage the underlying algebraic structure to deliver tractable complexity and improved error-correction performance.

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Algebraic Geometry Codes over Finite Fields publication trend

The graph below shows the total number of articles in algebraic geometry codes over finite fields across all publications each year (not limited to Nature Index journals).

Technical terms

Algebraic curve: A one-dimensional projective variety over a finite field, serving as the geometric object from which codes are constructed.

Finite field (Fq): A field with a finite number q of elements, providing the alphabet for code symbols and the base for the curve’s definition.

Divisor: A formal finite sum of points on a curve, whose degree controls the dimension of the Riemann–Roch space and hence the code’s dimension.

Riemann–Roch space: The vector space of functions on the curve whose poles and zeroes are bounded by a given divisor, used to define codewords by evaluation.

Weierstrass semigroup: The set of pole orders at a rational point on a curve, whose structure governs the gaps and pure gaps that influence code minimum distance.

Minimum distance: The smallest Hamming distance between distinct codewords, determining the error-correcting capability of the code.

References

1. Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties. Mathematics (2023).
2. Two-point AG codes from the Beelen-Montanucci maximal curve. Finite Fields and Their Applications (2022).
3. AG codes from Fq7-rational points of the GK maximal curve. Applicable Algebra in Engineering, Communication and Computing (2021).