  A new N-dimensional vector space of DNA sequences over the Galois field of
the 64 codons (GF(64)) was recently presented. Now, in order to include
deletions and insertions (indel mutations), we have defined a new Galois field
over the set of elements X1X2X3 (C125), where Xi belong to {O, A, C, G, C}. We
have called this set, the extended triplet set and the elements X1X2X3, the
extended triplets. The order of the bases is derived from the Z64-algebra of
the genetic code -recently published-. Starting from the natural bijection phi:
GF(5^3)-> C125 between the polynomial representation of elements from GF(5^3)
and the elements X1X2X3, a novel Galois field over the set of elements X1X2X3
is defined. Taking the polynomial coefficients a0, a1, a2 belong to GF(5) and
the bijective function f: GF(5) ->{O, A, C, G, C}, where f(0) = O, f(1) = A,
f(2) = C, f(3) = G, f(4) = U, bijection phi is induced such that phi(a0 + a1x +
a2x^2) = (f(a1), f(a2), f(a0)) = (X1X2X3). Next, by means of the bijection phi
we define sum "+" and product "*" operations in the set of codons C125, in such
a way that the resultant field (C125, +, *) turns isomorphic to the Galois
Field GF(5^3). This field allows the definition of a novel N-dimensional vector
space (S) over the field GF (5^3) on the set of all 125^N sequences of extended
triplets in which all possible DNA sequence alignments of length N are
included. Here the "classical gap" produced by alignment algorithms corresponds
to the neutral element "O". It is verified that the homologous (generalized)
recombination between two homologous DNA duplexes involving a reciprocal
exchange of DNA sequences -e.g. between two chromosomes that carry the same
genetic loci- algebraically corresponds to the action of two automorphism pairs
(or two translation pairs) over two paired DNA duplexes.
