Classification of some cosets of the Reed-Muller code (Q6088857)

Reed-Muller codes form a very classical family of error correcting codes, which have been used for more than 60 years. They also have strong links with the theory of cryptographic Boolean functions, since they can be seen as evaluation codes of these, under a bounded-degree limitation. The particular problem tackled in this nice paper is the classification of Boolean functions in 7 variables under the action of the affine general linear group. This action is crucial for cryptographic applications, because codes in the same orbits share their cryptographic properties (such as weak and strong differential uniformity). Unfortunately, the orbit number is huge and exhaustive classification is beyond reach for the foreseeable future. In this paper some ingenious computational methods are provided that let the authors arrive at the computation of RM(4,7)/RM(2,7) and of RM(7,7)/RM(3,7), which provide significant information: the former provides the complete classification of near bent functions in seven variables, the latter provides the value of the covering radius of R M(3, 7) (which was known, but with a totally different approach).