alternant code:

Let $ \alpha=\{\alpha_1,...,\alpha_n\}$ be a set of distinct elements of $ \mathbb{F}_{q^m}$ and let $ h=\{h_1,...,h_n\}$ be a set of non-zero elements of $ \mathbb{F}_{q^m}$. Fix a vector space basis for $ \mathbb{F}_{q^m}$ over $ \mathbb{F}_q$. For $ x\in \mathbb{F}_{q^m}$, let $ [x]$ denote the column vector representation of $ x$ with respect to this basis. Let

\begin{displaymath} H'= \left( \begin{array}{cccc} \ [h_1] & [h_2] & \dots &... ...2^{r-1}] & \dots & [h_n\alpha_n^{r-1}] \end{array} \right). \end{displaymath}
Assume $ r<n$. Delete any linearly dependent rows. Call the resulting matrix $ H$.

An alternant code $ {\cal A}(\alpha,h)$ is a code over $ \mathbb{F}_q$ with parity check matrix of the form $ H$, as above.