Celebratio Mathematica — Berlekamp

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E. R. Berlekamp and L. R. Welch: “Weight distributions of the cosets of the \( (32,6) \) Reed–Muller code,” IEEE Trans. Information Theory 18 : 1 (January 1972), pp. 203–207. MR 0396054 Zbl 0228.94003 article

In this paper we present the weight distribution of all \( 2^2\,6 \) cosets of the \( (32,6) \) first-order Reed–Muller code. The code is invariant under the complete affine group, of order \[ 32 \times 31 \times 30 \times 28 \times 24 \times 16 .\] In the Appendix we show (by hand computations) that this group partitions the \( 2^2\,6 \) cosets into only 48 equivalence classes, and we obtain the number of cosets in each class. A simple computer program then enumerated the weights of the 32 vectors in each of the 48 cosets. These coset enumerations also answer this equivalent problem: how well are the \( 2^3\,2 \) Boolean functions of five variables approximated by the \( 2^5 \) linear functions and their complements?