Computing the Weight Distribution of the Binary Reed-Muller Code ${\mathcal R} (4,9)$

TL;DR

This paper determines the weight distribution of the binary Reed-Muller code ${\mathcal R}(4,9)$ by integrating classical and recent classification techniques of Boolean functions and forms.

Contribution

It introduces a refined method combining historical and recent classifications to compute the weight distribution of a complex Reed-Muller code.

Findings

Computed the weight distribution of ${\mathcal R}(4,9)$

Integrated classical and modern Boolean function classifications

Provided insights into the structure of Reed-Muller codes

Abstract

We compute the weight distribution of the ${\mathcal R} (4,9)$ by combining the approach described in D. V. Sarwate's Ph.D. thesis from 1973 with knowledge on the affine equivalence classification of Boolean functions. To solve this problem posed, e.g., in the MacWilliams and Sloane book [p. 447], we apply a refined approach based on the classification of Boolean quartic forms in $8$ variables due to Ph. Langevin and G. Leander, and recent results on the classification of the quotient space ${\mathcal R} (4,7)/{\mathcal R} (2,7)$ due to V. Gillot and Ph. Langevin.