nLab linear code

Idea

In coding theory, a linear code is an error correcting code which is linear, in that it is a linear subspace of a vector space over a finite field $\mathbb{F}_q$. Specifically a binary linear code is a linear code over $\mathbb{F}_2$.

A linear code is a linear subspace of a vector space of finite dimension $d$ over a prime field $\mathbb{F}_p$, i.e. a vector space isomorphic to $(\mathbb{F}_p)^d$, for some prime number $p$ (often $p = 2$). The dimension $d$ of the vector space is also called the length of the linear code.

Examples

• Hamming code

Related concepts

• binary Golay code

• binary linear code

• Mathieu group

• combinatorial design

• Leech lattice

• adinkra

• quantum error correcting code

References

• Patrick Morandi, Error Correcting Codes and Algebraic Curves , lecture notes New Mexico State University 2001. (pdf)

• Jay A. Wood, Spinor groups and algebraic coding theory , J.Combinatorial Th. Series A 51 (1989) pp.277-313. (available online)

See also:

• Wikipedia, Linear code