quality:corruption, structural:reformatting
Simple encoding procedure: The message as a sequence of coefficients

In the original construction of Reed & Solomon (1960), the message m = (m 0, …, m k − 1) ∈ F k {\displaystyle m=(m_{0},\dots,m_{k-1})\in F^{k}} is mapped to the polynomial p m {\displaystyle p_{m}} with p m (a) = ∑ i = 0 k − 1 m i a i. {\displaystyle p_{m}(a)=\sum _{i=0}^{k-1}m_{i}a^{i}\,.} The codeword of m {\displaystyle m} is obtained by evaluating p m {\displaystyle p_{m}} at n {\displaystyle n} different points a 0, …, a n − 1 {\displaystyle a_{0},\dots,a_{n-1}} of the field F {\displaystyle F}. Thus the classical encoding function C: F k → F n {\displaystyle C:F^{k}\to F^{n}} for the Reed–Solomon code is defined as follows: C (m) = {\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}} This function C {\displaystyle C} is a linear mapping, that is, it satisfies C (m) = A m {\displaystyle C(m)=Am} for the following n × k {\displaystyle n\times k} -matrix A {\displaystyle A} with elements from F {\displaystyle F}: C (m) = A m = {\displaystyle 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}}}

This matrix is a Vandermonde matrix over F {\displaystyle F}. In other words, the Reed–Solomon code is a linear code, and in the classical encoding procedure, its generator matrix is A {\displaystyle A}.