In this paper we use the Gray code representation of the genetic code C=00, U=10, G=11 and A=01 (C pairs with G, A pairs with U) to generate a sequence of genetic code-based matrices. In connection with these code-based matrices, we use the Hamming distance to generate a sequence of numerical matrices. We then further investigate the properties of the numerical matrices and show that they are doubly stochastic and symmetric. We determine the frequency distributions of the Hamming distances, building blocks of the matrices, decomposition and iterations of matrices. We present an explicit decomposition formula for the genetic code-based matrix in terms of permutation matrices, which provides a hypercube representation of the genetic code. It is also observed that there is a Hamiltonian cycle in a genetic code-based hypercube.