Theoretical aspects of a design method for programmable NMR voters

We have designed a method for programmable NMR (N-modular redundancy) voters that relies on a binary matrix and enables easy scaling of the design regarding the number of voter inputs N. Thus, an automated construction of NMR systems is possible, given the basic module and arbitrary redundancy N. In this paper we present the mathematical aspects of the method, i.e., we analyze the properties of the matrix that characterizes the method. We give the characteristic polynomials of the properly and erroneously built matrices in their explicit forms. We further give the eigenvalues and corresponding eigenvectors of a properly built matrix and discuss the eigenvalues of an erroneously built matrix. We also give relations between the voter outputs and eigenpairs.