Weight enumerator for second-order Reed-Muller codes

In this paper, we establish the following result. Theorem: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_i</tex> , the number of codewords of weight <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> in the second-order binary Reed-Muller code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^m</tex> is given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_i = 0</tex> unless <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i = 2ᵐ⁻¹</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2ᵐ⁻¹ ± 2ᵐ⁻ˡ⁻ʲ</tex> , for some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j, 0 ≤ j ≤ [m/2], A_0 = A2^m = 1</tex> , and {equation} {split} A_{2ᵐ⁻¹ ± 2ᵐ⁻¹⁻ʲ} = 2ʲ⁽ʲ⁺¹⁾ &\{{(2^m - 1) (2ᵐ⁻¹ - 1 )}{4-1} \} \\ .&\{{(2ᵐ⁻² - 1)(2ᵐ⁻³ -1)}{4^2 - 1} \} ⋯ \\ .&\{{(2ᵐ⁻²ʲ⁺² -1)(2ᵐ⁻²ʲ⁺¹ -1)}{4^j -1} \} , \\ & 1 ≤ j ≤ [m/2] \\ {split} {equation} {equation} A_{2ᵐ⁻¹} = 2 \{ 2m(m+1)/2 - ∑ⱼ₌₀[m/2] A_{2ᵐ⁻¹ - 2ᵐ⁻¹⁻ʲ} \}. {equation}