The weight distribution of the fourth-order Reed–Muller code of length 512

Abstract We compute the weight distribution of the binary Reed–Muller code $${\mathcal {R}} (4,9)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>9</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> by combining the methodology described in D. V. Sarwate’s Ph.D. thesis from 1973 with newer results on the affine equivalence classification of Boolean functions. More specifically, to address this problem posed, e.g., in the book of MacWilliams and Sloane, we apply an enhanced approach based on the classification of Boolean quartic forms in eight variables due to Ph. Langevin and G. Leander, and the recent results on classification of the quotient space $${\mathcal {R}} (4,7)/{\mathcal {R}} (2,7)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>R</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> obtained by V. Gillot and Ph. Langevin.