Generator matrix - Citizendia

In coding theory, a generator matrix is a basis for a linear code, generating all its possible codewords. Coding theory is one of the most important and direct applications of Information theory. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics and Information theory, a linear code is an important type of Block code used in Error correction and detection schemes If the matrix is G and the linear code is C,

w=cG

where w is a unique codeword of the linear code C, c is a unique row vector, and a bijection exists between w and c. In Telecommunication, a code word is an element of a Code. Each code word is a Sequence of symbols assembled in accordance with the specific rules of In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property A generator matrix for a ( $n$ , $M = q k$ , $d$ )_$q$-code is of dimension $k * n$ . Here $n$ is the length of a codeword, $k$ is the number of information bits, $d$ is the minimum distance of the code, and $q$ is the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc. ). Note that the number of redundant bits is denoted $r = n - k$ . Redundancy in Information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message

The standard form for a generator matrix is

$G = \begin{bmatrix} I_k | P \end{bmatrix}$

where $I k$ is a $k * k$ identity matrix and P is of dimension $k * r$ . In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main

A generator matrix can be used to construct the parity check matrix for a code (and vice-versa). In Coding theory, a parity-check matrix of a linear block code C is a Generator matrix of the Dual code.

Equivalent Codes

Codes C₁ and C₂ are equivalent (denoted C₁ ~ C₂) if one code can be created from the other via the following two transformations:

permute components, and
scale components.

Equivalent codes have the same distance.

The generator matrices of equivalent codes can be obtained from one another via the following transformations:

permute rows
scale rows
add rows
permute columns, and
scale columns.

External links

MathWorld entry

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