Error-correcting properties of the SOLiD Exact Call Chemistry

BMC Bioinformatics

Table 1 Different representations of GF₄

Representation	Elements				⊕	0	1	α	β	⊗	1	α	β
GF₄	0	1	α	β	0	0	1	α	β	0	0	0	0
Nucleotides	A	C	G	T	1	1	0	β	α	1	1	α	β
Colors	0	1	2	3	α	α	β	0	1	α	α	β	1
					β	β	α	1	0	β	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, α⊕β=1 and α⊗α=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.

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