Alternant codes

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

$$H'= \left( \begin{array}{cccc} \ [h_1] & [h_2] & \dots &... ...2^{r-1}] & \dots & [h_n\alpha_n^{r-1}] \end{array} \right).$$

Assume $r<n$. Delete any linearly dependent rows. Call the resulting matrix $H$.

Definition 6.1.1   An alternant code ${\cal A}(\alpha,h)$ is a code over $\mathbb{F}_q$ with parity check matrix of the form $H$, as above.

Exercise 6.1.2   Compute all the vectors in an alternant code of your choosing. Verify the following properties.

• $n-mr\leq {\rm dim} {\cal A}(\alpha,h)\leq n-r$,
• the minimum distance of ${\cal A}(\alpha,h)$ is at least $r+1$,
• the length of ${\cal A}(\alpha,h)$ is $n$.