RESUMO
In this paper, we identify all possible Gray Code and Partitioned Gray Code representations of the Universal Genetic Code for n = 2-bit and 3-bit binary numbers. We analyse the Hamming Distance matrices of all these Gray code and Partitioned Gray Code possibilities for which we obtain the Toeplitz and Partitioned Toeplitz Matrices, respectively. We use this Gray Code and Partitioned Gray Code representations of the Universal Genetic Code combined with the novel Toeplitz matrix approach to generate many Never Born Protein (NBP) Sequences, which exhibit intrinsic structural stability. In general, Never Born Protein sequences may have many potential applications in synthetic biology and opens a new vista in understanding this new subset of proteins for better applications in drug discovery, synthesis of fine chemicals, etc.