Generalized Srivastava code[1]

Description

An \([n,k \geq n-mst,d \geq st+1 ]_q\) alternant code defined for \(n+s\) distinct elements \(\alpha_1,\alpha_2,\cdots,\alpha_n,w_1,w_2,\cdots,w_s\) and \(n\) nonzero elements \(z_1,z_2,\cdots,z_n\) of \(\mathbb{F}_{q^m}\).

The code’s parity-check matrix is [2; pg. 358] \begin{align} H=\begin{pmatrix} H_{1}\\ H_{2}\\ \vdots\\ H_{s} \end{pmatrix}~, \tag*{(1)}\end{align} where, for \(l=1,\ldots,s\), \begin{align} H_{l}=\begin{pmatrix} \frac{z_{1}}{\alpha_{1}-w_{l}} & \frac{z_{2}}{\alpha_{2}-w_{l}} & \cdots & \frac{z_{n}}{\alpha_{n}-w_{l}}\\ \frac{z_{1}}{\left(\alpha_{1}-w_{l}\right)^{2}} & \frac{z_{2}}{\left(\alpha_{2}-w_{l}\right)^{2}} & \cdots & \frac{z_{n}}{\left(\alpha_{n}-w_{l}\right)^{2}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{z_{1}}{\left(\alpha_{1}-w_{l}\right)^{t}} & \frac{z_{2}}{\left(\alpha_{2}-w_{l}\right)^{t}} & \cdots & \frac{z_{n}}{\left(\alpha_{n}-w_{l}\right)^{t}} \end{pmatrix}~. \tag*{(2)}\end{align}