Minimal codewords in Reed-Muller codes

Minimal codewords were introduced by Massey [8] for cryptographical purposes. They are used in particular secret sharing schemes, to model the access structures. We study minimal codewords of weight smaller than 3·2m−r in binary Reed-Muller codes RM(r, m) and translate our problem into a geometrical one, using a classification result of Kasami, Tokura, and Azumi [5, 6] on Boolean functions. In this geometrical setting, we calculate numbers of non-minimal codewords. So we obtain the number of minimal codewords in the cases where we have information about the weight distribution of the code RM(r, m). The presented results improve previous results obtained theoretically by Borissov, Manev, and Nikova [3], and computer aided results of Borissov and Manev [2]. This paper is in fact an extended abstract. Full proofs can be found on the arXiv.