Strong Trinucleotide Circular Codes

Table 1

(a) The first column contains the permutations 𝜎 of the symmetric group Σ₂ on the alphabet {𝑎_[2]}={𝑎₀,𝑎₁} and the second column contains their codes 𝑟(𝜎). (b) The first column contains the permutations 𝜎 of the symmetric group Σ₃ on the alphabet {𝑎_[3]}={𝑎₀,𝑎₁,𝑎₂} and the second column their codes 𝑟(𝜎). (c) The first column contains the permutations 𝜎 of the symmetric group Σ₄ on the alphabet {𝑎_[4]}={𝑎₀,𝑎₁,𝑎₂,𝑎₃} and the second column contains their codes 𝑟(𝜎). This table easily allows to determine the codes for permutations on any other totally ordered four-letter alphabet, in particular the alphabet [4]={0,1,2,3} (0<1<2<3), the genetic alphabet 𝒜₄ (𝐴<𝐶<𝐺<𝑇) and the alphabet {𝑎,𝑏,𝑐,𝑑} (𝑎<𝑏<𝑐<𝑑). For example, 211 is the code for 2130 on the alphabet [4], for 𝐺𝐶𝑇𝐴 on the alphabet 𝒜₄ and for 𝑐𝑏𝑑𝑎 on the alphabet {𝑎,𝑏,𝑐,𝑑}.

(a)

Permutation 𝜎	Code 𝑟(𝜎)
𝑎0𝑎1	0
𝑎1𝑎0	1

(b)

Permutation 𝜎	Code 𝑟(𝜎)
𝑎0𝑎1𝑎2	00
𝑎0𝑎2𝑎1	01
𝑎1𝑎0𝑎2	10
𝑎1𝑎2𝑎0	11
𝑎2𝑎0𝑎1	20
𝑎2𝑎1𝑎0	21

(c)

Permutation 𝜎	Code 𝑟(𝜎)
𝑎0𝑎1𝑎2𝑎3	000
𝑎0𝑎1𝑎3𝑎2	001
𝑎0𝑎2𝑎1𝑎3	010
𝑎0𝑎2𝑎3𝑎1	011
𝑎0𝑎3𝑎1𝑎2	020
𝑎0𝑎3𝑎2𝑎1	021
𝑎1𝑎0𝑎2𝑎3	100
𝑎1𝑎0𝑎3𝑎2	101
𝑎1𝑎2𝑎0𝑎3	110
𝑎1𝑎2𝑎3𝑎0	111
𝑎1𝑎3𝑎0𝑎2	120
𝑎1𝑎3𝑎2𝑎0	121
𝑎2𝑎0𝑎1𝑎3	200
𝑎2𝑎0𝑎3𝑎1	201
𝑎2𝑎1𝑎0𝑎3	210
𝑎2𝑎1𝑎3𝑎0	211
𝑎2𝑎3𝑎0𝑎1	220
𝑎2𝑎3𝑎1𝑎0	221
𝑎3𝑎0𝑎1𝑎2	300
𝑎3𝑎0𝑎2𝑎1	301
𝑎3𝑎1𝑎0𝑎2	310
𝑎3𝑎1𝑎2𝑎0	311
𝑎3𝑎2𝑎0𝑎1	320
𝑎3𝑎2𝑎1𝑎0	321