The Algebra of Genomes – From Galois Fields to Random Walks.

Resumen: Motivation: By encoding nucleotide sequences as elements of finite algebraic structures, rather than as strings over a fixed alphabet, we obtain representations that are simultaneously more compact and more computationally tractable, enabling direct algebraic operations on genomic data without decompression. 

Main Results: We present explicit constructions showing that key sequence-analysis tasks (alignment, motif-finding, compression) admit natural formulations in this algebraic setting, with preliminary results suggesting substantial gains in both storage and query performance over standard formats. 

Open Questions: We close with conjectural directions linking these structures to deeper symmetries of the genetic code itself, raising the question of whether the algebra is not merely a convenient encoding but reflects an underlying mathematical structure of biological information.