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Weight enumerator for second-order Reed-Muller codes

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2 Author(s)

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}

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Information Theory, IEEE Transactions on  (Volume:16 ,  Issue: 6 )