Weight distributions of the cosets of the (32,6) Reed-Muller code

In this paper we present the weight distribution of all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^26</tex> cosets of the (32,6) first-order Reed-Muller code. The code is invariant under the complete affine group, of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">32 × 31 × 30 × 28 × 24 ×</tex> 16. In the Appendix we show (by hand computations) that this group partitions the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^26</tex> 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 ih each of the 48 cosets. These coset enumerations also answer this equivalent problem: how well are the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^32</tex> Boolean functions of five variables approximated by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^5</tex> linear functions and their complements?