= Generalized Reed-Solomon code
{title2=$\operatorname{GRS}_k(a,v)$}
= GRS code
{c}
{synonym}
For $1\le k\le n\le q$, choose distinct evaluation points $a_1,\ldots,a_n$ in a <finite field> $\mathbb F_q$ and nonzero coordinate multipliers $v_1,\ldots,v_n$. The <generalized Reed-Solomon code> is
$$
\operatorname{GRS}_k(a,v)=\{(v_i f(a_i))_{i=1}^{n}:\deg f<k\}.
$$
Multiplying 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 $\operatorname{GRS}_{n-k}(a,u)$, where one may take
$$
u_i=v_i^{-1}\prod_{j\ne i}(a_i-a_j)^{-1}.
$$
A 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>.
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