  A new n-dimensional vector space of the DNA sequences on the Galois field of
the 64 codons (GF(64)) is proposed. In this vector space gene mutations can be
considered linear transformations or translations of the wild type gene. In
particular, the set of translations that preserve the chemical type of the
third base position in the codon is a subgroup which describes the most
frequent mutations observed in mutational variants of four genes: human
phenylalanine hydroxylase (PAH), human beta globin (HBG), HIV-1 Protease (HIVP)
and HIV-1 Reverse transcriptase (HIVRT). Furthermore, an inner pseudo-product
defined between codons tends to have a positive value when the codons code to
similar amino acids and a negative value when the codons code to amino acids
with extreme hydrophobic properties. Consequently, it is found that the inner
pseudo-product between the wild type and the mutant codons tends to have a
positive value in the mutational variants of the genes: PAH, HBG, HIVP, HIVRT.
