On binary codes related to mutually quasi-unbiased weighing matrices

Some mutually quasi-unbiased weighing matrices are constructed from binary codes satisfying the conditions that the number of non-zero weights of the code is four and the code contains the first order Reed–Muller code. Motivated by this, in this note, we study binary codes satisfying the conditions. The weight distributions of binary codes satisfying the conditions are determined. We also give a classification of binary codes of lengths 8, 16 and binary maximal codes of length 32 satisfying the conditions. As an application, sets of 8 mutually quasi-unbiased weighing matrices for parameters (16, 16, 4, 64) and 4 mutually quasi-unbiased weighing matrices for parameters (32, 32, 4, 256) are constructed for the first time.