Weight enumerator for second-order Reed-Muller codes

In this paper, we establish the following result. Theorem:
A_i
, the number of codewords of weight
i
in the second-order binary Reed-Muller code of length
2^m
is given by
A_i = 0
unless
i = 2^{m-1}
or
2^{m-1} \pm 2^{m-l-j}
, for some
j, 0 \leq j \leq [m/2], A_0 = A_{2^m} = 1
, and \begin{equation} \begin{split} A_{2^{m-1} \pm 2^{m-1-j}} = 2^{j(j+1)} &\{\frac{(2^m - 1) (2^{m-1} - 1 )}{4-1} \} \\ .&\{\frac{(2^{m-2} - 1)(2^{m-3} -1)}{4^2 - 1} \} \cdots \\ .&\{\frac{(2^{m-2j+2} -1)(2^{m-2j+1} -1)}{4^j -1} \} , \\ & 1 \leq j \leq [m/2] \\ \end{split} \end{equation} \begin{equation} A_{2^{m-1}} = 2 \{ 2^{m(m+1)/2} - \sum_{j=0}^{[m/2]} A_{2^{m-1} - 2^{m-1-j}} \}. \end{equation}