Write Generator Matrix (2,4) of Reed Muller code of (2,4)

Question

I was wondering how to go about this sum, since I wasnt able to figure out how to solve this. Could any one help me out on this?

Answer 1

The second order Reed-Muller code of length $2^4$ is of dimension $1 + 4 + \binom{4}{2} = 11$. Denote the data bits as $\mathbf a = (a_0, a_1, a_2, a_3, a_4, a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, a_{34})$, and define the function $f(x_1,x_2, x_3, x_4)$ of $4$ Boolean variables as $$\begin{align}f(x_1, x_2, x_3, x_4) &= a_0 \oplus a_1x_1 \oplus a_2x_2 \oplus a_3x_3 \oplus a_4x_4 \\ &\qquad \oplus a_{12}x_1x_2 \oplus a_{13}x_1x_3 \oplus a_{14}x_1x_4 \oplus a_{23}x_2x_3 \oplus a_{24}x_2x_4 \oplus a_{34}x_3x_4. \end{align}$$ The codeword corresponding to the data bits $a_0, a_1, a_2, a_3, a_4, a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, a_{34}$ is $$\mathbf c = \bigr(f(0,0,0,0), \quad f(0,0,0,1), \quad f(0,0,1,0),\quad f(0,0,1,1), \quad f(0,1,0,0), \quad \ldots, \quad f(1,1,1,1)\bigr)$$ Thus, the codeword is $\mathbf c = \mathbf aG$ where the $11$ rows of $G$ are the codewords corresponding to the functions $1, x_1, x_2, x_3, x_4, x_1x_2, x_1x_3, x_1x_4, x_2x_3, x_2x_4, x_3x_4$.