Hadamard square of linear codes and the generalized minimal distance of Reed–Muller code of order 2

Abstract We propose a new technique for dimensional analysis of the Hadamard (Schur) square of an error-correcting linear code. This is usually achieved by a representation of the Hadamard square as an image of some linear operator defined on the set of quadratic forms. A link between the dimension of the Hadamard square the rank of some submatrix of the generating matrix of the code containing the set of vector values of quadratic forms is established. So the dimensional analysis of the Hadamard square can be carried out with the extensive code-based machinery rather than via the approach with estimation of the number of joint zeros of the set of quadratic forms. As a result and we establish a nonasymptotic estimate for the probability that the Hadamard square of a random linear code fills the entire space. This estimate can be used for cryptographic analysis of post-quantum code-based cryptosystems.