On some cosets of the first-order Reed-Muller code with highminimum weight

We study a family of particular cosets of the first-order Reed-Muller code R(1,m): those generated by special codewords, the idempotents. Thus we obtain new maximal weight distributions of cosets of R(1,7) and 84 distinct almost maximal weight distributions of cosets of R(1,9), that is, with minimum weight 240. This leads to crypotographic applications in the context of stream ciphers