Weight enumerator for second-order Reed-Muller codes

In this paper, we establish the following result. Theorem: , the number of codewords of weight in the second-order binary Reed-Muller code of length is given by unless or , for some , 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}