Editing Reed–Solomon error correction (section)

==== Simple encoding procedure: The message as a sequence of coefficients ====
In the original construction of Reed and Solomon, the message <math>m=(m_0,\dots,m_{k-1}) \in F^k
</math> is mapped to the polynomial <math>p_m</math> with
<math display="block">p_m(a) = \sum_{i=0}^{k-1} m_i a^{i} \,.</math>
The codeword of <math>m</math> is obtained by evaluating <math>p_m</math> at <math>n</math> different points <math>a_0, \dots, a_{n-1}</math> of the field <math>F</math>.<ref name="ReedSolomon"/> Thus the classical encoding function <math>C:F^k \to F^n</math> for the Reed–Solomon code is defined as follows:
<math display="block">C(m) = \begin{bmatrix}
p_m(a_0) \\
p_m(a_1) \\
\cdots \\
p_m(a_{n-1})
\end{bmatrix}</math>
This function <math>C</math> is a [[linear mapping]], that is, it satisfies <math>C(m) = Am
</math> for the following <math>n \times k</math>-matrix <math>A</math> with elements from <math>F</math>:
<math display="block">C(m) = Am = \begin{bmatrix}
1 & a_0 & a_0^2 & \dots & a_0^{k-1} \\
1 & a_1 & a_1^2 & \dots & a_1^{k-1} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & a_{n-1} & a_{n-1}^2 & \dots  & a_{n-1}^{k-1}
\end{bmatrix}
\begin{bmatrix}
m_0 \\
m_1 \\
\vdots \\
m_{k-1}
\end{bmatrix}
</math>

This matrix is a [[Vandermonde matrix]] over <math>F</math>. In other words, the Reed–Solomon code is a [[linear code]], and in the classical encoding procedure, its [[Linear code#Properties|generator matrix]] is <math>A</math>.