Posted by: atri | February 6, 2009

## Lecture 11: Explicit codes and Reed-Solmon codes

In today’s lecture, we formally defined what we mean by explicit (and strongly explicit) codes. Then we defined Reed-Solomon codes and saw that it is a linear code. Towards the end of the class, I just mentioned  the generator matrix of the Reed-Solomon code without “justification.” Let me give more details on that now.

Say the message polynomial is $P(X)=\sum_{i=0}^{k-1} m_iX^i$ which is evaluated over the set of points $\alpha_1,\dots,\alpha_n$. Note that this implies that the $j$th entry in the codeword is $\sum_{i=0}^{k-1} m_i (\alpha_j)^i$. In other words,

$RS(m_0,\cdots,m_{k-1})=(m_0,\cdots,m_{k-1})\cdot\left(\begin{array} {ccccc} 1&\cdots&1&\cdots&1\\\alpha_1&\cdots&\alpha_j&\cdots&\alpha_n\\\vdots&&\vdots&&\vdots\\\alpha_1^i&\cdots&\alpha_j^i&\cdots&\alpha_n^i\\ \vdots&&\vdots&&\vdots\\\alpha_1^{k-1}&\cdots&\alpha_j^{k-1}&\cdots&\alpha_n^{k-1}\end{array}\right)$.

Thus, the generator matrix of the Reed-Solomon code as its $j$th column is the vector $(1,\alpha_j,\cdots,\alpha_j^{k-1})$. This kind of matrix is called the Vandermonde matrix.

For more details on the material covered in class today, see the fall notes on lecture 12.