Algebraic geometry codes, often abbreviated AG codes, are a type of linear code that generalize Reed–Solomon codes. The Russian mathematician V. D. Goppa...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...

of general type Zariski surface Algebraic variety Hypersurface Quadric (algebraic geometry) Dimension of an algebraic variety Hilbert's Nullstellensatz...

This is a list of algebraic coding theory topics....

geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry...

Networks. AN codes Algebraic geometry code BCH code, which can be designed to correct any arbitrary number of errors per code block. Barker code used for...

Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the algebraic group of square roots...

codes, of which BCH codes are an example Reed–Solomon codes Reed–Muller code Algebraic geometry code Binary Goppa code Low-density parity-check codes...

Bose–Chaudhuri–Hochquenghem, Turbo, Golay, algebraic geometry codes, low-density parity-check codes, and space–time codes. Error detecting codes can be optimised to detect...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

where the properties of codes are expressed in algebraic terms and then further researched.[citation needed] Algebraic coding theory is basically divided...

methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or...

sets, or finite geometries. On the algebraic side, besides group theory and representation theory, lattice theory and commutative algebra are commonly used...

representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory[citation needed], combinatorics and number theory. Within...

relative distance for codes over alphabets of size less than 49.[citation needed] For larger alphabets, algebraic geometry codes sometimes achieve an asymptotically...

Reed–Solomon and Algebraic-Geometry Codes" introducing an algorithm that allowed for the correction of errors beyond half the minimum distance of the code. It applies...

important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic...

2024-01-27. Danilov, V. I. (2006). "II. Algebraic Varieties and Schemes". Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes. Springer....

operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually...

for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics...

a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma...

distance of the code. It applies to Reed–Solomon codes and more generally to algebraic geometry codes. This algorithm produces a list of codewords (it...

binary Goppa codes (subfield codes of algebraic geometry codes of a genus-0 curve over finite fields of characteristic 2); these codes can be efficiently...

early geometry. (See Areas of mathematics and Algebraic geometry.) The earliest recorded beginnings of geometry can be traced to early peoples, such as the...

relation between algebraic geometry and codes, utilizing the Riemann-Roch theorem. Today these codes are called algebraic geometry codes. In 1981 he presented...

theory and linear algebra taught in a first degree course, as well as a little projective geometry, and a very little algebraic geometry." When q is a prime...

(algebraic surfaces) Proper base change theorem (algebraic geometry) Puiseux's theorem (algebraic geometry) Ramanujam vanishing theorem (algebraic geometry)...

arithmetic coding techniques 1977 – Abraham Lempel and Jacob Ziv develop Lempel–Ziv compression (LZ77) 1982 – Valerii Denisovich Goppa introduces algebraic geometry...

equations Valery Goppa, inventor of Goppa codes, and algebraic geometry codes in the field of algebraic geometry Mikhail Gromov, a prominent developer of...

continuous deformations. Algebraic topology, the use in topology of algebraic methods, mainly homological algebra. Discrete geometry, the study of finite...