A report on Coding theory

A two-dimensional visualisation of the Hamming distance, a critical measure in coding theory.

Study of the properties of codes and their respective fitness for specific applications.

- Coding theory
A two-dimensional visualisation of the Hamming distance, a critical measure in coding theory.

15 related topics with Alpha

Overall

A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction.

Information theory

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Scientific study of the quantification, storage, and communication of digital information.

Scientific study of the quantification, storage, and communication of digital information.

A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction.

Channel coding is concerned with finding such nearly optimal codes that can be used to transmit data over a noisy channel with a small coding error at a rate near the channel capacity.

To clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied Reed–Solomon error correction (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted.

Error detection and correction

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To clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied Reed–Solomon error correction (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted.

In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communication channels.

Comparison of spectrograms of audio in an uncompressed format and several lossy formats. The lossy spectrograms show bandlimiting of higher frequencies, a common technique associated with lossy audio compression.

Data compression

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Process of encoding information using fewer bits than the original representation.

Process of encoding information using fewer bits than the original representation.

Comparison of spectrograms of audio in an uncompressed format and several lossy formats. The lossy spectrograms show bandlimiting of higher frequencies, a common technique associated with lossy audio compression.
Solidyne 922: The world's first commercial audio bit compression sound card for PC, 1990
Processing stages of a typical video encoder

Other topics associated with compression include coding theory and statistical inference.

Hamming distance

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Number of positions at which the corresponding symbols are different.

Number of positions at which the corresponding symbols are different.

A major application is in coding theory, more specifically to block codes, in which the equal-length strings are vectors over a finite field.

Polynomial code

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In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length) that are divisible by a given fixed polynomial (of shorter length, called the generator polynomial).

Data communication

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Transfer and reception of data in the form of a digital bitstream or a digitized analog signal over a point-to-point or point-to-multipoint communication channel.

Transfer and reception of data in the form of a digital bitstream or a digitized analog signal over a point-to-point or point-to-multipoint communication channel.

The theoretical aspects of data transmission are covered by information theory and coding theory.

If 00010111 is a valid codeword, applying a right circular shift gives the string 10001011. If the code is cyclic, then 10001011 is again a valid codeword. In general, applying a right circular shift moves the least significant bit (LSB) to the leftmost position, so that it becomes the most significant bit (MSB); the other positions are shifted by 1 to the right.

Cyclic code

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If 00010111 is a valid codeword, applying a right circular shift gives the string 10001011. If the code is cyclic, then 10001011 is again a valid codeword. In general, applying a right circular shift moves the least significant bit (LSB) to the leftmost position, so that it becomes the most significant bit (MSB); the other positions are shifted by 1 to the right.

In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code.

BCH code

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In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called Galois field).

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Mathematics

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Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Coding theory, including error correcting codes and a part of cryptography

Repetition code

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In coding theory, the repetition code is one of the most basic error-correcting codes.