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In mathematics, a (binary) linear code of length and rank is a linear subspace C with dimension k of the vector space Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
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Aside: is the field of two elements and is the set of all n-tuples of length n over . Occasionally some other finite field containing q > 2 elements is used, in which case the code is said to be a "q-ary" code (rather than a binary code). Special exceptions to the general adjective "q-ary" are binary and ternary codes (corresponding to q=2 and q=3 respectively). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...
In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...
In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...
An adjective is a part of speech which modifies a noun, usually making its meaning more specific. ...
Properties
By virtue of the fact that the code is a subspace of , the sum c1 + c2 of two codewords in C is also a codeword (ie an element of the subspace C). Thus the entire code (which may be very large) to be represented as the span of a minimal set of codewords (known as a basis in linear algebra terms). These basis codewords are often collated in the rows of a matrix known as a generating matrix for the code C. In mathematics, if a set with certain properties is called a space, then a subset of which with same properties is usually called a subspace. ...
In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...
In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. ...
In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
The subspace definition also gives rise to the important property that the minimum Hamming distance between codewords is simply the minimum Hamming weight of all codewords since: In information theory, the Hamming distance, named after Richard Hamming, is the number of positions in two strings of equal length for which the corresponding elements are different. ...
The Hamming weight of a string of bits is the number of 1s in it. ...
implying: The motivation behind creating linear codes is to allow for syndrome decoding.
Popular notation Codes in general are often denoted by the letter C. A linear code of length n, rank k (ie having k codewords in its basis and k rows in its generating matrix) and minimum Hamming weight d is referred to as an (n,k,d) code. CODE is a visual programming language and system for parallel programming, letting users compose sequential programs into parallel ones. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
Remark. This is not to be confused with the notation [n,r,d] to denote a non-linear code of length n, size r (ie having r codewords) and minimum Hamming distance d.
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