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

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

Minimal Codewords in Linear Codes

2000 Mathematics Subject Classification: 94B05, 94B15.Cyclic binary codes C of block length n = 2^m − 1 and generator polynomial g(x) = m1(x)m2^s+1(x), (s, m) = 1, are considered. The cardinalities of the sets of minimal codewords of weights 10 and 11 in codes C and of weight 12 in their extended codes ^C are determined. The weight distributions of minimal codewords in the binary Reed-Muller codes RM (3, 6) and RM (3, 7) are determined. The applied method enables codes with larger parameters to be attacked