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

For $1\le k\le n\le q$, choose distinct evaluation points $a_1,\dots ,a_n$ in a finite field ${\mathbb{F}}_q$ and nonzero coordinate multipliers $v_1,\dots ,v_n$. The generalized Reed-Solomon code isMultiplying coordinates by nonzero field elements preserves Hamming weight, so this is a linear code with parameters $[n,k,n-k+1]$, just as for a Reed-Solomon code. In particular it is a maximum distance separable code.

Its dual code is ${\mathrm{GRS}}_{n-k}(a,u)$, where one may takeA common nonzero multiplier on all $u_i$ gives the same dual code. For $k=n$ the dual code is the zero code, interpreted as having no nonzero evaluation polynomials.