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A [[code]] of fixed length $n$ over a finite field $F$ which forms a subspace of the vector space $F^n$. 

A linear code of rank $r$ may be represented by a ''generator matrix'', an $(r \times n)$ matrix whose rows form a set of linearly independent code words.  A generator matrix can be put in the form $G = (I_r | G_1)$ by row operations, showing that a linear code is necessarily a [[systematic code]].  The components of the code word corresponding to the columns of $G_1$ may be referred to as check digits.

A linear code $C$ has a '''dual code''' $C^\perp$ consisting of all vectors in $F^n$ orthogonal to every element of $C$ with respect to the bilinear form $(x,y) = \sum_{i=1}^n x_i y_i$.  This is a linear code of rank $(n-r)$, and a basis for $C^\perp$ is a set of ''parity check'' vectors: a generator matrix for $C^\perp$ is a '''parity check matrix'''.

See [[Error-correcting code]].

====References====
* Goldie, Charles M.; Pinch, Richard G.E. ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) ISBN 0-521-40456-8 {{ZBL|0746.94001}}
* van Lint, J.H., "Introduction to coding theory" (2nd ed.) Graduate Texts in Mathematics '''86''' Springer (1992) ISBN 3-540-54894-7 {{ZBL|0747.94018}}