Minimal codewords in Reed–Muller codes

Abstract

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 3·2 m−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.

On the minimum number of minimal codewords

“…However, the method is highly inefficient when the size of the code is large. For some linear codes, the minimal codewords were determined by exploiting special properties of those codes, for instance see [1,7,9,12,23,27].…”

Construction of minimal linear codes from multi-variable functions

“…In addition, to reduce the complexity of a decoding algorithm, the authors in [1,15] took account of the set of minimal codewords in C. Thus it has been an interesting research topic in coding theory whether or not a codeword in C is minimal. To find all minimal codewords of C is however very hard for general linear codes such as binary Reed-Muller codes [1,5,6,18].…”

On the non-minimality of the largest weight codewords in the binary Reed-Muller codes

“…In two articles [3,4], the codewords of RM(r, m) of weight smaller than 5d/2 = 2 m−r+1 + 2 m−r−1 are classified. In particular, the codewords of weight smaller than 2d are the incidence vectors of (m − r)-dimensional subspaces of AG(m, 2), particular quadrics of AG(m, 2) and of symmetric differences of (m − r)-dimensional subspaces of AG(m, 2) [3,8].…”

“…By [3,8], every codeword c in RM(r, m) of weight smaller than 2d corresponds to the incidence vector of an (m − r)-dimensional subspace of AG(m, 2), a particular quadric of AG(m, 2) or to a symmetric difference of two (m − r)-dimensional affine subspaces of AG(m, 2). This enabled Schillewaert, Storme, and Thas to improve the results of Borissov, Manev, and Nikova by counting the number of non-minimal codewords of RM(r, m) of every weight in RM(r, m) smaller than 3d.…”

“…This enabled Schillewaert, Storme, and Thas to improve the results of Borissov, Manev, and Nikova by counting the number of non-minimal codewords of RM(r, m) of every weight in RM(r, m) smaller than 3d. For the exact formula of the number of non-minimal codewords of a weight smaller than 3d in RM(r, m), we refer to [8].…”