Generalized Reed-Solomon code (${\mathrm{GRS}}_k(a,v)$)

A generalized Reed-Solomon code consists of $(v_{i}f(a_i){)}_{i=1}^n$, where the evaluation points are distinct, all coordinate multipliers $v_i$ are nonzero, and $\mathrm{deg}f<k$. It remains a maximum distance separable code. Its dual code is another generalized Reed-Solomon code with dimension $n-k$ and multipliers proportional to $v_i^{-1}{\prod }_{j\ne i}(a_i-a_j{)}^{-1}$.