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In this paper, we establish the following result. Theorem: Aᵢ, the number of codewords of weight i in the second-order binary Reed-Muller code of length 2ᵐ is given by Aᵢ = 0 unless i = 2^m-1 or 2^m-1 2^m-l-j, for some j, 0 j m/2, A₀ = A₂㵯 = 1, and equation split A₂^₌-₁ 2^{m-1-j} = 2^j (j+1) &\ (2ᵐ - 1) (2^{m-1 - 1) 4-1 \} \\. &\ (2^{m-2 - 1) (2^{m-3 -1) }4² - 1 \} \\. &\ (2^{m-2j+2 -1) (2^{m-2j+1 -1) }4ʲ -1 \}, \\ & 1 j m/2 \\ split equation equation A₂^₌-₁ = 2 \ 2^{m (m+1) /2 - ₉=₀^m/2 A₂^₌-₁ - 2^{m-1-j} \}. equation
Sloane et al. (Sun,) studied this question.