Interleaved RS (IRS) code

Description

A modification of RS codes where multiple polynomials are used to define each codeword.

Each codeword \(\mu\) of a \(t\)-interleaved RS code is a string of values of the corresponding set \(\{f_\mu^{(1)},f_\mu^{(2)},\cdots,f_\mu^{(t)}\}\) of \(t\) polynomials at the points \(\alpha_i\). The vector codewords can be arranged in an array whose rows are ordinary RS codes for each polynomial \(f_\mu^{(j)}\), yielding the encoding \begin{align} \mu\to\left( \begin{array}{cccc} f_{\mu}^{(1)}\left(\alpha_{1}\right) & f_{\mu}^{(1)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(1)}\left(\alpha_{n}\right)\\ f_{\mu}^{(2)}\left(\alpha_{1}\right) & f_{\mu}^{(2)}\left(\alpha_{2}\right) & & f_{\mu}^{(2)}\left(\alpha_{n}\right)\\ \vdots & & \ddots & \vdots\\ f_{\mu}^{(t)}\left(\alpha_{1}\right) & f_{\mu}^{(t)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(t)}\left(\alpha_{n}\right) \end{array}\right)~.\tag*{$(1)$} \end{align}