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Mathematical linear code
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_code
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Soviet and Russian mathematician
relation between algebraic geometry and codes, utilizing the Riemann-Roch theorem. Today these codes are called algebraic geometry codes. In 1981 he presented
Valery_Goppa
Scheme for controlling errors in data over noisy communication channels
} . 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
Error_correction_code
of general type Zariski surface Algebraic variety Hypersurface Quadric (algebraic geometry) Dimension of an algebraic variety Hilbert's Nullstellensatz
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Branch of mathematics
methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or
Geometry
is a list of algebraic coding theory topics. ARQ Adler-32 Algebraic geometry code BCH code BCJR algorithm Belief propagation Berger code Berlekamp–Massey
List of algebraic coding theory topics
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
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Algebra based on a vector space with a quadratic form
Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the algebraic group of square roots
Clifford_algebra
Algorithm on linear-feedback shift registers
some algebraic geometry codes, including one-point algebraic geometry codes, and variants have been developed for multipoint codes from algebraic curves
Berlekamp–Massey_algorithm
System of rules to convert information into another form or representation
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
Code
relative distance for codes over alphabets of size less than 49.[citation needed] For larger alphabets, algebraic geometry codes sometimes achieve an asymptotically
Gilbert–Varshamov bound for linear codes
Gilbert–Varshamov_bound_for_linear_codes
Area of combinatorics
sets, or finite geometries. On the algebraic side, besides group theory and representation theory, lattice theory and commutative algebra are commonly used
Algebraic_combinatorics
Study of the properties of codes and their fitness
where the properties of codes are expressed in algebraic terms and then further researched.[citation needed] Algebraic coding theory is basically divided
Coding_theory
Branch of mathematics
2024-01-27. Danilov, V. I. (2006). "II. Algebraic Varieties and Schemes". Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes. Springer.
Algebra
Quantum error correction code
overcome this difficulty. Algebraic-geometry codes provide another source of stabilizer constructions. Matsumoto used algebraic curves to obtain asymptotically
Stabilizer_code
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
Studies linear representations of finite groups over fields of positive characteristic
representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory[citation needed], combinatorics and number theory. Within
Modular_representation_theory
Branch of functional analysis
operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually
Operator_algebra
Error-correcting codes
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
Reed–Solomon_error_correction
of geometry. Fundamentally, it studies algebraic varieties. Algebraic graph theory a branch of graph theory in which methods are taken from algebra and
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Historical development of geometry
early geometry. (See Areas of mathematics and Algebraic geometry.) The earliest recorded beginnings of geometry can be traced to early peoples, such as the
History_of_geometry
Indian computer scientist (born 1976)
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
Venkatesan_Guruswami
arithmetic coding techniques 1977 – Abraham Lempel and Jacob Ziv develop Lempel–Ziv compression (LZ77) 1982 – Valerii Denisovich Goppa introduces algebraic geometry
Timeline of information theory
Timeline_of_information_theory
Computer algebra system
for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics
SageMath
Class of error-correcting code
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
Linear_code
Australian mathematician
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
James William Peter Hirschfeld
James_William_Peter_Hirschfeld
(algebraic surfaces) Proper base change theorem (algebraic geometry) Puiseux's theorem (algebraic geometry) Ramanujam vanishing theorem (algebraic geometry)
List_of_theorems
Computer system for solving algebra problems
a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma
Magma (computer algebra system)
Magma_(computer_algebra_system)
theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory,
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
equations Valery Goppa, inventor of Goppa codes, and algebraic geometry codes in the field of algebraic geometry Mikhail Gromov, a prominent developer of
List of Russian mathematicians
List_of_Russian_mathematicians
Russian mathematician
the American Mathematical Society in 2012. Codes on Algebraic Curves, Kluwer 1999 Arithmetic of Algebraic Curves, New York, Plenum Publishing 1994, Russian
Sergei Stepanov (mathematician)
Sergei_Stepanov_(mathematician)
Relation between genus, degree, and dimension of function spaces over surfaces
important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic
Riemann–Roch_theorem
Mathematics study in geometry
mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
gonality of an algebraic curve C is defined as the lowest degree of a nonconstant rational map from C to the projective line. In more algebraic terms, if C
Gonality of an algebraic curve
Gonality_of_an_algebraic_curve
Classification scheme for mathematics
polynomials 13: Commutative algebra (Commutative rings and algebras) 14: Algebraic geometry 15: Linear and multilinear algebra; matrix theory 16: Associative
Mathematics Subject Classification
Mathematics_Subject_Classification
Mathematical construct in computer algebra
and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind
Gröbner_basis
Topics referred to by the same term
elliptic partial differential equations Regular algebra, or Kleene algebra Regular code, an algebraic code with a uniform distribution of distances between
Regular
Study of discrete mathematical structures
programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete
Discrete_mathematics
Awarded every year by the American Mathematical Society
Robin (1975). "Equivalence relations on algebraic cycles and subvarieties of small codimension". Algebraic Geometry – Arcata 1974. Proceedings of Symposia
Leroy_P._Steele_Prize
Interactive geometry software (IGS) or dynamic geometry environments (DGEs) are computer programs which allow one to create and then manipulate geometric
List of interactive geometry software
List_of_interactive_geometry_software
Topics referred to by the same term
Norwegian power company The Bernstein–Khovanskii–Kushnirenko theorem, in algebraic geometry Betriebskrankenkasse (de), a type of insurance for healthcare in Austria
BKK_(disambiguation)
Algebraic structure with addition, multiplication, and division
Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly
Field_(mathematics)
Asymmetric encryption algorithm developed by Robert McEliece
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
McEliece_cryptosystem
Basic concepts of algebra
relationships in science and mathematics are expressed as algebraic equations. In mathematics, a basic algebraic operation is a mathematical operation similar to
Elementary_algebra
Field of knowledge
continuous deformations. Algebraic topology, the use in topology of algebraic methods, mainly homological algebra. Discrete geometry, the study of finite
Mathematics
Topics referred to by the same term
and cryptography. Construction of the fiber product of schemes, in algebraic geometry. Change of basis Base change (disambiguation) This disambiguation
Change_of_base
of geometry Glossary of differential geometry and topology Glossary of Riemannian and metric geometry Glossary of scheme theory List of algebraic geometry
Lists_of_mathematics_topics
Branch of pure mathematics
(for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in
Number_theory
Mathematics award
Bhargav Bhatt – "For outstanding work in commutative algebra and arithmetic algebraic geometry, particularly on the development of p-adic cohomology
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Mathematical manifold theory
theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is intrinsically
Hodge_theory
Topics referred to by the same term
Dijkstra's algorithm Formal spectrum of a ring, a construction in algebraic geometry Superplastic forming, of sheet metal Spray polyurethane foam, a building
SPF
monotonous and sometimes problematic algebraic manipulation tasks. The primary difference between a computer algebra system and a traditional calculator
List of open-source software for mathematics
List_of_open-source_software_for_mathematics
Mathematical problem
for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ2 related to the Routh–Hurwitz theorem: all of
Dividing a square into similar rectangles
Dividing_a_square_into_similar_rectangles
British-Russian mathematician
algebraic geometry. His work has focused on rational points, the Hasse principle, the Manin obstruction, exponential sums, and error-correcting codes
Alexei_Skorobogatov
Algebraic structure
computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. A finite field is a field that
Finite_field
Stéphane; Perret, Marc; Zaytsev, Alexey (eds.). Algorithmic Arithmetic, Geometry, and Coding Theory. Contemporary Mathematics. Vol. 637. Providence, RI: American
Geometric_class_field_theory
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
Algebraic manipulation of "true" and "false"
connection between his algebra and logic was later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many other
Boolean_algebra
Number with a real and an imaginary part
complex numbers is defined as the (unique) algebraic extension field of the real numbers later in #Abstract algebraic definitions. The solution in radicals
Complex_number
Topics referred to by the same term
Control-alt-delete Cylindrical algebraic decomposition, a notion and an algorithm in computer algebra and real algebraic geometry A right-continuous function
Cad_(disambiguation)
Geometric concept of a 2D space with "points at infinity" adjoined
example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP2
Projective_plane
Software used in mathematical applications
Mathematica Z3 Theorem Prover Archimedes Geo3D Cabri Geometry CaRMetal Cinderella Dr. Geo GeoGebra Geometry Expert GEUP Kig KSEG The Geometer's Sketchpad TracenPoche
Mathematical_software
Branch of mathematics that studies abstract algebraic structures
representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix
Representation_theory
Set of n^3 + 1 points arranged into subsets of n + 1
In geometry, a unital is a set of n 3 + 1 {\displaystyle n^{3}+1} points arranged into subsets of size n + 1 so that every pair of distinct points of the
Unital_(geometry)
Branch of discrete mathematics
algebra. Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods
Combinatorics
Austrian mathematician
Austrian mathematician known for his work in discrepancy theory, algebraic geometry, quasi-Monte Carlo methods, and cryptography. Niederreiter was born
Harald_Niederreiter
Interactive geometry, algebra and calculus application
GeoGebra (a portmanteau of geometry and algebra) is an interactive geometry, algebra, statistics and calculus application, intended for learning and teaching
GeoGebra
Qualification in mathematics study
Elementary Mathematics, with additional topics including Algebra binomial expansion, proofs in plane geometry, differential calculus and integral calculus. Additional
Additional_Mathematics
Topics referred to by the same term
used in computational geometry RELAX NG, an XML schema language whose files use the extension .rng Rng (algebra), an algebraic structure similar to rings
RNG
Calculus of vector-valued functions
in geometric algebra, as described below. The algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined
Vector_calculus
Algebraic operation on coordinate vectors
algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry
Dot_product
Area of mathematics
techniques in natural languages Computational algebraic geometry Computational group theory Computational geometry Computational number theory Computational
Computational_mathematics
American mathematician
applications from and to noncommutative algebraic geometry, as well as earlier work in random matrix theory, coding theory, lattices, and quantum information
Eric_M._Rains
Python library for symbolic computation
Taylor series (Laurent series) Systems of linear equations Systems of algebraic equations that are solvable by radicals Differential equations Difference
SymPy
In algebra, expression of an ideal as the intersection of ideals of a specific type
Noetherian rings. The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union
Primary_decomposition
Mathematical behavior near singularities
of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As
Monodromy
Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable
Euler_calculus
of curves and surfaces with algebraic representation. Franco P. Preparata; Michael Ian Shamos (1985). Computational Geometry - An Introduction. Springer-Verlag
List of books in computational geometry
List_of_books_in_computational_geometry
absolutely irreducible if it is irreducible over every algebraic extension of K, and an affine algebraic set defined by equations with coefficients in a field
Absolute_irreducibility
Russian mathematician
April 16, 1972) is a mathematician, known for his works in algebraic topology, algebraic geometry and mathematical physics. Barannikov graduated with honors
Serguei_Barannikov
Associative algebra together with a Lie bracket that satisfies Leibniz's law
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also
Poisson_algebra
connection on a vector bundle over a family of algebraic varieties Gauss–Newton line – described in Journal for Geometry and Graphics, see also Newton line Gauss's
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Branch of computer science
Computational geometry is a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry. Some purely geometrical
Computational_geometry
Johnson–Lindenstrauss lemma (Euclidean geometry) Margulis lemma Lebesgue's number lemma (dimension theory) Gauss's lemma (Riemannian geometry) Craig interpolation lemma
List_of_lemmas
Mathematics independent of applications
number theory, where these infinities are typically countable and algebraic geometry where functions are typically tamed functions (i.e. piecewise polynomial
Pure_mathematics
development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam, and the development of an algebraic notation by al-Qalasādī
History_of_mathematics
Topological model
and exterior (E) of geometries a and b. The terms interior and boundary in this article are used in the sense used in algebraic topology and manifold
DE-9IM
complex geometry, spin manifolds, the Dirac operator, and algebraic cycles Ruth I. Michler (1967–2000), American commutative algebraist and algebraic geometer
List_of_women_in_mathematics
Geometry with 7 points and 7 lines
and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study
Fano_plane
Upper bound in coding theory
1053661 Vermani, L. R. (1996), Elements of algebraic coding theory, Chapman & Hall Welsh, Dominic (1988), Codes and Cryptography, Oxford University Press
Singleton_bound
American mathematician (1937–2006)
Speaker at the ICM in Nice with the talk Transcendence and differential algebraic geometry. In the 1970s, he worked on the fundamentals of physics, including
James_Ax
Type of block code
another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detection and
Cyclic_code
Infinitesimal calculus on functions defined on a geometric algebra
theories including vector calculus, differential geometry, and differential forms. With a geometric algebra given, let a {\displaystyle a} and b {\displaystyle
Geometric_calculus
alternative is to use the optional argument algebraic, then the formula can be described as an algebraic expression. pstricks-add extends pst-plot enabling
PSTricks
2008 British TV series or programme
André Weil who built algebraic geometry, a whole new language. Weil's work connected number theory, algebra, topology and geometry. Finally du Sautoy mentions
The_Story_of_Maths
Counting polynomial roots in an interval
with algebraic numbers. For computing with algebraic numbers, a common method is to represent them as a pair of a polynomial to which the algebraic number
Sturm's_theorem
Vector graphics using a relative cursor on a Cartesian plane
coordinate models, including non-Euclidean geometry, are allowed but not included. The following Python code uses the turtle module to create a rainbow
Turtle_graphics
Class of mathematical software
differential geometry calculations GRTensorM is a computer algebra package for performing calculations in the general area of differential geometry. MathGR
Tensor_software
French mathematician
mathematician. He made contributions in the fields of abstract algebra, algebraic geometry, and computer vision, and participated in the Nicolas Bourbaki
Michel_Demazure
travel, tourism, insurance
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
ALGEBRAIC GEOMETRY-CODE
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