Table 1

Different representations of GF4
Representation Elements 0 1 α β 0 1 α β
GF4 0 1 α β 0 0 1 α β 0 0 0 0 0
Nucleotides A C G T 1 1 0 β α 1 0 1 α β
Colors 0 1 2 3 α α β 0 1 α 0 α β 1
β β α 1 0 β 0 β 1 α

For the purposes of the coding theory presented here, both nucleotides and colors represent elements of the Galois Field over four elements (GF4) and the correspondence between them is shown below. For example, the color ‘2’, the nucleotide ‘G’ and the element ‘α’ are considered to be equivalent for the purposes of calculation. A field consists of a set of elements and rules on how to add (⊕) and multiply (⊗) them together; the results of combining two elements are expressed by the Cayley tables above; for example, αβ=1 and αα=β. The standard rules for associativity and commutativity for multiplication and addition still apply in finite fields, and multiplication is still distributive over addition [8]. One notable difference from ordinary arithmetic is that all elements are self-invertible under addition in GF4, so addition and subtraction are equivalent operations.

Massingham and Goldman

Massingham and Goldman BMC Bioinformatics 2012 13:145   doi:10.1186/1471-2105-13-145

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