# Linear code

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**.

#### 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

**How to Cite This Entry:**

Linear code.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Linear_code&oldid=39150