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.

Systems of elements of genetic code are studied by their cognitive presentation in a form of mathematical matrices of symbolic and numerical kinds. This cognitive form of data presentation permits to discover new phenomenological rules of evolution of genetic codes, to reveal two branches of evolution within genetic code, to present hidden interrelations between the golden section and parameters of genetic polyplets, to disclose matrices of a hyperbolic turn in genetic matrices, etc. Mysterious sets of structures, realized by the nature in a hierarchi al system of genetic codes, can be confronted by a heuristic manner with families of mathematical matrices, which contain elements of these structures. A few rules of degeneracy and segregations of genetic codes are revealed in this direction. A new answer on the fundamental question-"why 20 amino acids?"-is proposed as well. .

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