Let
V be a vector space over
Z/
pZ
with
p a prime.
Definition
A
code in
V
is a set of vectors in
V. The vectors of a code are called
code words.
A
linear
code in
V is a linear subspace of
V. If
C
is a linear code of dimension
k in the
n-dimensional
vector space
V, then
C is referred to as an (
n,k)
code.
In the world of digital communication, the
binary number system is used a lot nowadays. Therefore we confine
ourselves here mainly to codes in vector spaces over Z/2Z. In these
vector spaces, scalar multiplication is very simple: there are only
two scalars, 0 and 1. These codes are known as
binary
codes.
Definition
Let
C be a code in
V. The
distance
between two vectors from
V is the number of coordinate positions at
which the two vectors differ. The
minimal
distance of
C is the minimum taken over all distances between any
two different code words from
C.
A code is a useful code if the
length of the code is small and the minimal distance is large. In the
remainder of this section we will describe a method to construct useful
error-correcting codes with the help of polynomials.