Linear code: Difference between revisions

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== Generator and check matrices ==

As a [[linear subspace]] of <math>\mathbb{F}_q^n</math>, the entire code ''C'' (which may be very large) may be represented as the [[span (linear algebra)|span]] of a set of <math>nk</math> codewords (known as a [[basis (linear algebra)|basis]] in [[linear algebra]]). These basis codewords are often collated in the rows of a matrix G known as a '''[[Generator matrix|generating matrix]]''' for the code ''C''. When G has the block matrix form <math>\boldsymbol{G} = [I_k | P]</math>, where <math>I_k</math> denotes the <math>k \times k</math> identity matrix and P is some <math>k \times (n-k)</math> matrix, then we say G is in '''standard form'''.

A matrix ''H'' representing a linear function <math>\phi : \mathbb{F}_q^n\to \mathbb{F}_q^{n-k}</math> whose [[Kernel (matrix)|kernel]] is ''C'' is called a '''[[check matrix]]''' of ''C'' (or sometimes a parity check matrix). Equivalently, ''H'' is a matrix whose [[null space]] is ''C''. If ''C'' is a code with a generating matrix ''G'' in standard form, <math>\boldsymbol{G} = [I_k | P]</math>, then <math>\boldsymbol{H} = [-P^{T} | I_{n-k} ]</math> is a check matrix for C. The code generated by ''H'' is called the '''dual code''' of C. It can be verified that G is a <math>k \times n</math> matrix, while H is a <math>(n-k) \times n</math> matrix.