Strong Trinucleotide Circular Codes

Abstract

Recently, we identified a hierarchy relation between trinucleotide comma-free codes and trinucleotide circular codes (see our previous works). Here, we extend our hierarchy with two new classes of codes, called 𝐷𝐿𝐷 and 𝐿𝐷𝐿 codes, which are stronger than the comma-free codes. We also prove that no circular code with 20 trinucleotides is a 𝐷𝐿𝐷 code and that a circular code with 20 trinucleotides is comma-free if and only if it is a 𝐿𝐷𝐿 code. Finally, we point out the possible role of the symmetric group ∑₄ in the mathematical study of trinucleotide circular codes.

1. Introduction

We continue our study of the combinatorial properties of trinucleotide circular codes. A trinucleotide is a word of three letters (triletter) on the genetic alphabet {𝐴,𝐶,𝐺,𝑇}. The set of 64 trinucleotides is a code in the sense of language theory, more precisely a uniform code but not a circular code (Remark 2.4 and [1, 2]). In order to have an intuitive meaning of these notions, codes are written on a straight line while circular codes are written on a circle, but, in both cases, unique decipherability is required. Circular codes are some particular subsets of the 64 trinucleotide set while comma-free codes are even more constrained subsets.

In the past 50 years, comma-free codes and circular codes have been studied in theoretical biology, mainly to understand the structure and the origin of the genetic code as well as the reading frame (construction) of genes, for example [3–5]. Before the discovery of the genetic code, Crick et al. [3] proposed a (maximal) comma-free code of 20 trinucleotides for coding the 20 amino acids. In 1996, a (maximal) circular code 𝑋₀ of 20 trinucleotides was identified statistically on two large and different gene populations, eukaryotes, and prokaryotes [6]. During the last years, circular codes are mathematical objects studied in discrete mathematics, theoretical computer science, and theoretical biology, for example [7–22]. In particular, in theory of codes, there are some unexpected common notions between variable length circular codes and trinucleotide circular codes [17, 19, 21, 22].

Recently, we proposed a hierarchy relation between the trinucleotide comma-free codes and the trinucleotide circular codes (Proposition 3 in [23]). More precisely, all the trinucleotide codes in this hierarchy are circular, the strongest ones being comma-free. In this paper, we identify two new classes of trinucleotide circular codes which are stronger than the comma-free codes.

We introduce here the following new notions. A set 𝑋 of trinucleotides has the property 𝐷𝐿𝐷 if for any trinucleotides 𝑡,𝑡^′∈𝑋, no letter occurs both as a proper suffix of 𝑡 and a proper prefix of 𝑡^′. A set 𝑋 of trinucleotides has the property 𝐿𝐷𝐿 if for any trinucleotides 𝑡,𝑡^′∈𝑋, no diletter occurs both as a proper suffix of 𝑡 and a proper prefix of 𝑡^′. These sets 𝐷𝐿𝐷 and 𝐿𝐷𝐿 are not only trinucleotides circular codes but they are also stronger than the comma-free codes (Propositions 3.4 and 3.5, and Remarks 3.6 and 3.7). We also prove that no circular code with 20 trinucleotides is a 𝐷𝐿𝐷 code (Proposition 3.10) and that a circular code with 20 trinucleotides is comma-free if and only if it is a 𝐿𝐷𝐿 code (Proposition 3.11).

Therefore, our previous hierarchy (Proposition 3 in [23] recalled in Proposition 2.17 below) is extended with these new 𝐷𝐿𝐷 and 𝐿𝐷𝐿 classes of strong trinucleotides circular codes (Proposition 4.1).

Finally, a curious relation with the symmetric group Σ₄ appears again. The tables given here and the other symmetric relations identified previously (e.g., Proposition 6 in [23]) suggest that the symmetric group Σ₄ can play an important role in the mathematical study of these trinucleotide circular codes. However, we have no formal mathematical explanation so far.

2. Preliminaries

Let 𝒜 denote a finite alphabet, 𝒜^∗ the free monoid over 𝒜 and 𝒜⁺ the free semigroup over 𝒜. The elements of 𝒜^∗ are words and the empty word, denoted by 𝜀, is the identity of 𝒜^∗. Given a subset 𝑋 of 𝒜^∗, 𝑋^𝑛 is the set of the words over 𝒜 which are the products of 𝑛 words from 𝑋, that is, 𝑋^𝑛={𝑥₁𝑥₂⋯𝑥_𝑛∣𝑥_𝑖∈𝑋}. If 𝑋 is a (finite) set, then |𝑋| denotes its cardinality and if 𝑢 is a word, then |𝑢| denotes its length. A word 𝑢 is a factor of a word 𝑣 if there exist two words 𝑢^′ and 𝑢^′′ such that 𝑣=𝑢^′𝑢𝑢^′′. When 𝑢^′=𝜀 (resp. 𝑢^′′=𝜀), 𝑢 is a prefix (resp. suffix) of 𝑣. A proper factor (resp. proper prefix, proper suffix) 𝑢 of 𝑣 is a factor (resp. prefix, suffix) 𝑢 of 𝑣 such that |𝑢|<|𝑣|.

There is a correspondence between the genetic and language-theoretic concepts. The letters (or nucleotides or bases) define the genetic alphabet 𝒜₄={𝐴,𝐶,𝐺,𝑇}. The set of nonempty words (resp. words) over 𝒜₄ is denoted by 𝒜⁺₄ (resp. 𝒜^∗₄). The set of the 16 words of length 2 (or dinucleotides or diletters) is denoted by 𝒜²₄. The set of the 64 words of length 3 (or trinucleotides or triletters) is denoted by 𝒜³₄. The total order over the alphabet 𝒜₄ is 𝐴<𝐶<𝐺<𝑇. Consequently, 𝒜⁺₄ is lexicographically ordered: given two words 𝑢,𝑣∈𝒜⁺₄, 𝑢 is smaller than 𝑣 in lexicographical order, written 𝑢<𝑣, if and only if either 𝑢 is a proper prefix of 𝑣 or there exist 𝑥,𝑦∈𝒜₄, 𝑥<𝑦, and 𝑟,𝑠,𝑡∈𝒜^∗₄ such that 𝑢=𝑟𝑥𝑠 and 𝑣=𝑟𝑦𝑡.

Definition 2.1. Code: a subset 𝑋 of 𝒜⁺ is a code over 𝒜 if for each 𝑥₁,…,𝑥_𝑛,𝑥^′₁,…,𝑥^′_𝑚∈𝑋, 𝑛,𝑚≥1, the condition 𝑥₁⋯𝑥_𝑛=𝑥^′₁⋯𝑥^′_𝑚 implies 𝑛=𝑚 and 𝑥_𝑖=𝑥^′_𝑖 for 𝑖=1,…,𝑛.

For any 𝑘-letter alphabet, 𝑘≥1, and for any word length 𝑙, 𝑙≥1, 𝒜^𝑙_𝑘 is a code. In particular, 𝒜³₄ is a code. More precisely, it is a uniform code [1]. Consequently, any nonempty subset of 𝒜³₄ is a code, called trinucleotide code in this paper.

Definition 2.2. Trinucleotide comma-free code: a trinucleotide code 𝑋⊂𝒜³₄ is comma-free if for each 𝑦∈𝑋 and 𝑢,𝑣∈𝒜^∗₄ such that 𝑢𝑦𝑣=𝑥₁⋯𝑥_𝑛 with 𝑥₁,…,𝑥_𝑛∈𝑋, 𝑛≥1, it results that 𝑢,𝑣∈𝑋^∗.

Several varieties of trinucleotide comma-free codes were described in [18].

Definition 2.3. Trinucleotide circular code: a trinucleotide code 𝑋⊂𝒜³₄ is circular if for each 𝑥₁,…,𝑥_𝑛,𝑥^′₁,…,𝑥^′_𝑚∈𝑋, 𝑛,𝑚≥1, 𝑝∈𝒜^∗₄, 𝑠∈𝒜⁺₄, the conditions 𝑠𝑥₂⋯𝑥_𝑛𝑝=𝑥^′₁⋯𝑥^′_𝑚 and 𝑥₁=𝑝𝑠 imply 𝑛=𝑚, 𝑝=𝜀 and 𝑥_𝑖=𝑥^′_𝑖 for 𝑖=1,…,𝑛.

Remark 2.4. 𝒜³₄ is obviously not a circular code and even less a comma-free code. However, several subsets of 𝒜³₄ are trinucleotide circular codes (e.g., Propositions 2.12 and 2.13).

Definition 2.5. Maximal trinucleotide circular code: a trinucleotide circular code 𝑋⊂𝒜³₄ is maximal if for each 𝑥∈𝒜³₄, 𝑥∉𝑋, 𝑋∪{𝑥} is not a trinucleotide circular code.

Definition 2.6. A trinucleotide circular code containing exactly 𝑘 elements is called a 𝑘-trinucleotide circular code.

Remark 2.7. A 20-trinucleotide circular code is(i)maximal (in the sense that it cannot be contained in a trinucleotide circular code with more words);(ii)maximum (in the sense that no trinucleotide circular code can contain more than 20 words).

We now recall some definitions and previous results related to the trinucleotide circular code necklaces. In the sequel, 𝑙₁,𝑙₂,…,𝑙_𝑛 are letters in 𝒜₄, 𝑑₁,𝑑₂,…,𝑑_𝑛 are diletters in 𝒜²₄, and 𝑛 is an integer satisfying 𝑛≥2.

Definition 2.8. Letter Diletter Necklaces (𝐿𝐷𝑁): we say that the ordered sequence 𝑙₁,𝑑₁,𝑙₂,𝑑₂,…,𝑑_𝑛−1,𝑙_𝑛,𝑑_𝑛 is an 𝑛𝐿𝐷𝑁 for a subset 𝑋⊂𝒜³₄ if 𝑙₁𝑑₁,𝑙₂𝑑₂,…,𝑙_𝑛𝑑_𝑛∈𝑋 and 𝑑₁𝑙₂,𝑑₂𝑙₃,…,𝑑_𝑛−1𝑙_𝑛∈𝑋.

Definition 2.9. Letter Diletter Continued Necklaces (𝐿𝐷𝐶𝑁): we say that the ordered sequence 𝑙₁,𝑑₁,𝑙₂,𝑑₂,…,𝑑_𝑛−1,𝑙_𝑛,𝑑_𝑛,𝑙_𝑛+1 is an (𝑛+1)𝐿𝐷𝐶𝑁 for a subset 𝑋⊂𝒜³₄ if 𝑙₁𝑑₁,𝑙₂𝑑₂,…,𝑙_𝑛𝑑_𝑛∈𝑋 and 𝑑₁𝑙₂,𝑑₂𝑙₃,…,𝑑_𝑛−1𝑙_𝑛,𝑑_𝑛𝑙_𝑛+1∈𝑋.

Definition 2.10. Diletter Letter Necklaces (𝐷𝐿𝑁): we say that the ordered sequence 𝑑₁,𝑙₁,𝑑₂,𝑙₂,…,𝑙_𝑛−1,𝑑_𝑛,𝑙_𝑛 is an 𝑛𝐷𝐿𝑁 for a subset 𝑋⊂𝒜³₄ if 𝑑₁𝑙₁,𝑑₂𝑙₂,…,𝑑_𝑛𝑙_𝑛∈𝑋 and 𝑙₁𝑑₂,𝑙₂𝑑₃,…,𝑙_𝑛−1𝑑_𝑛∈𝑋.

Definition 2.11. Diletter Letter Continued Necklaces (𝐷𝐿𝐶𝑁): we say that the ordered sequence 𝑑₁,𝑙₁,𝑑₂,𝑙₂,…,𝑙_𝑛−1,𝑑_𝑛,𝑙_𝑛,𝑑_𝑛+1 is an (𝑛+1)𝐷𝐿𝐶𝑁 for a subset 𝑋⊂𝒜³₄ if 𝑑₁𝑙₁,𝑑₂𝑙₂,…,𝑑_𝑛𝑙_𝑛∈𝑋 and 𝑙₁𝑑₂,𝑙₂𝑑₃,…,𝑙_𝑛−1𝑑_𝑛,𝑙_𝑛𝑑_𝑛+1∈𝑋.

Proposition 2.12 (see [17]). Let 𝑋 be a trinucleotide code. The following conditions are equivalent:(i)𝑋 is a circular code;(ii)𝑋 has no 5𝐿𝐷𝐶𝑁.

Proposition 2.13 (see [18]). Let 𝑋 be a trinucleotide code. The following conditions are equivalent:(i)𝑋 is a comma-free code.(ii)𝑋 has no 2𝐿𝐷𝑁 and no 2𝐷𝐿𝑁.

Definition 2.14. Let 𝑋 be a trinucleotide code. For any integer 𝑛∈{2,3,4,5}, we say that 𝑋 belongs to the class 𝐶^{𝑛𝐿𝐷𝑁} if 𝑋 has no 𝑛𝐿𝐷𝑁 and that 𝑋 belongs to the class 𝐶^{𝑛𝐷𝐿𝑁} if 𝑋 has no 𝑛𝐷𝐿𝑁. Similarly, for any integer 𝑛∈{3,4,5}, we say that 𝑋 belongs to the class 𝐶^{𝑛𝐿𝐷𝐶𝑁} if 𝑋 has no 𝑛𝐿𝐷𝐶𝑁 and that 𝑋 belongs to the class 𝐶^{𝑛𝐷𝐿𝐶𝑁} if 𝑋 has no 𝑛𝐷𝐿𝐶𝑁.

Notation 1. For any integer 𝑛∈{2,3,4,5}, 𝐼^𝑛=𝐶^{𝑛𝐿𝐷𝑁}∩𝐶^{𝑛𝐷𝐿𝑁} and 𝑈^𝑛=𝐶^{𝑛𝐿𝐷𝑁}∪𝐶^{𝑛𝐷𝐿𝑁}. Similarly, for any integer 𝑛∈{3,4,5}, 𝐼^𝑛𝐶=𝐶^{𝑛𝐿𝐷𝐶𝑁}∩𝐶^{𝑛𝐷𝐿𝐶𝑁} and 𝑈^𝑛𝐶=𝐶^{𝑛𝐿𝐷𝐶𝑁}∪𝐶^{𝑛𝐷𝐿𝐶𝑁}.

Proposition 2.15 (see [23]). The following chains of inclusions hold:(i)𝐶^{2𝐿𝐷𝑁}⊂𝐶^{3𝐿𝐷𝐶𝑁}⊂𝐶^{3𝐿𝐷𝑁}⊂𝐶^{4𝐿𝐷𝐶𝑁}⊂𝐶^{4𝐿𝐷𝑁}⊂𝐶^{5𝐿𝐷𝐶𝑁}⊂𝐶^{5𝐿𝐷𝑁}; (ii)𝐶^{2𝐷𝐿𝑁}⊂𝐶^{3𝐷𝐿𝐶𝑁}⊂𝐶^{3𝐷𝐿𝑁}⊂𝐶^{4𝐷𝐿𝐶𝑁}⊂𝐶^{4𝐷𝐿𝑁}⊂𝐶^{5𝐷𝐿𝐶𝑁}⊂𝐶^{5𝐷𝐿𝑁}; (iii)𝐶^{2𝐿𝐷𝑁}⊂𝐶^{3𝐷𝐿𝐶𝑁}⊂𝐶^{3𝐿𝐷𝑁}⊂𝐶^{4𝐷𝐿𝐶𝑁}⊂𝐶^{4𝐿𝐷𝑁}⊂𝐶^{5𝐷𝐿𝐶𝑁}⊂𝐶^{5𝐿𝐷𝑁}; (iv)𝐶^{2𝐷𝐿𝑁}⊂𝐶^{3𝐿𝐷𝐶𝑁}⊂𝐶^{3𝐷𝐿𝑁}⊂𝐶^{4𝐿𝐷𝐶𝑁}⊂𝐶^{4𝐷𝐿𝑁}⊂𝐶^{5𝐿𝐷𝐶𝑁}⊂𝐶^{5𝐷𝐿𝑁}; (v)𝐼²⊂𝐼³𝐶⊂𝐼³⊂𝐼⁴𝐶⊂𝐼⁴⊂𝐼⁵𝐶⊂𝐼⁵; (vi)𝑈²⊂𝑈³𝐶⊂𝑈³⊂𝑈⁴𝐶⊂𝑈⁴⊂𝑈⁵𝐶⊂𝑈⁵.

Remark 2.16. By Proposition 2.13, the chain of inclusions of Proposition 2.15 (𝑣) begins with 𝐼² which is the class of comma-free codes.

Proposition 2.17. With 20-trinucleotide circular codes, the following chains of inclusions and equalities hold: 𝐼²⊂𝑈²=𝐼³𝐶⊂𝑈³𝐶=𝐼³⊂𝑈³=𝐼⁴𝐶⊂𝑈⁴𝐶=𝐼⁴⊂𝑈⁴=𝐼⁵𝐶⊂𝑈⁵𝐶=𝐼⁵=𝑈⁵. (2.1)

3. Strong Trinucleotide Circular Codes

We introduce new definitions which impose very strong conditions on the words of a subset of 𝒜³₄. These word subsets, strongly constrained, are indeed new circular codes which are stronger than the trinucleotide comma-free codes according to the following propositions.

Definition 3.1. A subset 𝑋 of 𝒜³₄ has the 𝐷𝐿𝐷 property if, for any 𝑙₁,𝑙₂,𝑙₃,𝑙^′₁,𝑙^′₂,𝑙^′₃∈𝒜₄, the conditions 𝑙₁𝑙₂𝑙₃∈𝑋 and 𝑙^′₁𝑙^′₂𝑙^′₃∈𝑋 imply 𝑙₁≠𝑙^′₃.

No letter of 𝒜₄ can occur in the first position of a trinucleotide of 𝑋 when it is also in the last position of another trinucleotide of 𝑋.

Definition 3.2. A subset 𝑋 of 𝒜³₄ has the 𝐿𝐷𝐿 property if, for any 𝑙₁,𝑙^′₁∈𝒜₄, 𝑑₁,𝑑^′₁∈𝒜²₄, the conditions 𝑙₁𝑑₁∈𝑋 and 𝑑^′₁𝑙^′₁∈𝑋 imply 𝑑₁≠𝑑^′₁.

No diletter of 𝒜²₄ can occur as a prefix of a trinucleotide of 𝑋 when it is also a suffix of another trinucleotide of 𝑋.

Remark 3.3. The trinucleotide code {𝐴𝐶𝐺,𝐺𝑇𝐴} is not a 𝐷𝐿𝐷-strong trinucleotide circular code but it is a 𝐿𝐷𝐿-strong trinucleotide circular code. The trinucleotide code {𝐴𝐶𝐺,𝐶𝐺𝑇} is not a 𝐿𝐷𝐿-strong trinucleotide circular code but it is a 𝐷𝐿𝐷-strong trinucleotide circular code.

Therefore, the class of 𝐷𝐿𝐷-strong trinucleotide circular codes is different from the class of 𝐿𝐷𝐿-strong trinucleotide circular codes. However, both are very particular cases of comma-free codes according to the following propositions.

Proposition 3.4. A 𝐷𝐿𝐷-strong trinucleotide circular code over 𝒜₄ is comma-free.

Proof. Suppose that 𝑋 is a 𝐷𝐿𝐷-strong trinucleotide circular code and, by way of contradiction, that it is not comma-free. Then, there exist two trinucleotides 𝑥𝑦𝑧,𝑥^′𝑦^′𝑧^′∈𝑋 such that either 𝑦𝑧𝑥^′ or 𝑧𝑥^′𝑦^′ are in 𝑋. In the first case, 𝑥^′ is a prefix of 𝑥^′𝑦^′𝑧^′ and a suffix of 𝑦𝑧𝑥^′ while in the second case, 𝑧 is a prefix of 𝑧𝑥^′𝑦^′ and a suffix of 𝑥𝑦𝑧. In both cases, 𝑋 is not a 𝐷𝐿𝐷-strong circular code. This is a contradiction.

Proposition 3.5. A 𝐿𝐷𝐿-strong trinucleotide circular code over 𝒜₄ is comma-free.

Proof. Suppose that 𝑋 is a 𝐿𝐷𝐿-strong trinucleotide circular code and, by way of contradiction, that it is not comma-free. Then, there exist two trinucleotides 𝑥𝑦𝑧,𝑥^′𝑦^′𝑧^′∈𝑋 such that either 𝑦𝑧𝑥^′ or 𝑧𝑥^′𝑦^′ are in 𝑋. In the first case, 𝑦𝑧 is a prefix of 𝑦𝑧𝑥^′ and a suffix of 𝑥𝑦𝑧 while in the second case, 𝑥^′𝑦^′ is a prefix of 𝑥^′𝑦^′𝑧^′ and a suffix of 𝑧𝑥^′𝑦^′. In both cases, 𝑋 is not a 𝐿𝐷𝐿-strong circular code. This is a contradiction.

Remark 3.6. There are trinucleotide comma-free codes which are not 𝐷𝐿𝐷-strong trinucleotide circular codes. Example: {𝐴𝐶𝐴}.

Remark 3.7. There are trinucleotide comma-free codes which are not 𝐿𝐷𝐿-strong trinucleotide circular codes. Example: {𝐴𝐶𝐺,𝐶𝐺𝑇}.

The two following propositions are obvious.

Proposition 3.8. For any letters 𝑥,𝑦,𝑧∈𝒜₄, a trinucleotide singleton 𝑥𝑦𝑧∈𝒜³₄ is a 𝐷𝐿𝐷-strong trinucleotide circular code over 𝒜₄ if and only if 𝑥≠𝑧.

Proposition 3.9. For any letters 𝑥,𝑦,𝑧∈𝒜₄, a trinucleotide singleton 𝑥𝑦𝑧∈𝒜³₄ is a 𝐿𝐷𝐿-strong trinucleotide circular code over 𝒜₄ if and only if at least two of its letters are different.

Remark 3.3 showed that 𝐷𝐿𝐷-strong and 𝐿𝐷𝐿-strong trinucleotide circular codes are different classes. The following propositions give more information about their difference.

Proposition 3.10. No 20-trinucleotide circular code can be a 𝐷𝐿𝐷-strong trinucleotide circular code.

Proof. Suppose, by way of contradiction, that a 20-trinucleotide circular code 𝑋 is also a 𝐷𝐿𝐷-strong trinucleotide circular code. Let 𝑃 (resp. 𝑆) be the set containing the letters 𝑙₁ (resp. 𝑙₃) of the trinucleotides 𝑙₁𝑙₂𝑙₃ of 𝑋. We have 𝑃∩𝑆=∅ (otherwise, 𝑋 has not the 𝐷𝐿𝐷 property), |𝑃|>1 (otherwise, 𝑋 has at most 16 elements) and |𝑆|>1 (otherwise, 𝑋 has at most 16 elements). Using Pigeon Hole Principle, it follows that 𝒜₄ has two disjoint subsets, say {𝑎,𝑏} and {𝑐,𝑑}, such that 𝑃={𝑎,𝑏} and 𝑆={𝑐,𝑑}. Consequently, 𝑋 has at most the following elements: 𝑎𝐴𝑐,𝑎𝐶𝑐,𝑎𝐺𝑐,𝑎𝑇𝑐,𝑎𝐴𝑑,𝑎𝐶𝑑,𝑎𝐺𝑑,𝑎𝑇𝑑,𝑏𝐴𝑐,𝑏𝐶𝑐,𝑏𝐺𝑐,𝑏𝑇𝑐,𝑏𝐴𝑑,𝑏𝐶𝑑,𝑏𝐺𝑑,𝑏𝑇𝑑, so we have again at most 16 elements. This is a contradiction.

Proposition 3.11. A 20-trinucleotide circular code is comma-free if and only if it is a 𝐿𝐷𝐿-strong trinucleotide circular code.

Proof. If. By Proposition 3.5, any 𝐿𝐷𝐿-strong trinucleotide circular code 𝑋 is also comma-free.
Only if. Suppose that 𝑋 is comma-free and, by way of contradiction, that it is not a 𝐿𝐷𝐿-strong trinucleotide circular code. Then, there exist two letters 𝑎,𝑏∈𝒜₄ and a diletter 𝑑₁∈𝒜²₄ such that 𝑎𝑑₁,𝑑₁𝑏∈𝑋. As 𝑋 cannot contain two elements in the same conjugation class, the condition 𝑎≠𝑏 holds. So, 𝒜₄−{𝑎,𝑏} contains exactly two elements, say 𝑐 and 𝑑.
𝑋 being a comma-free code, 𝑋 must contain exactly one trinucleotide in each of the 20 conjugation classes. By considering the conjugation class {𝑎𝑎𝑐,𝑎𝑐𝑎,𝑐𝑎𝑎}, only 𝑎𝑎𝑐 can belong to 𝑋. Indeed, {𝑎𝑐𝑎,𝑎𝑑₁,𝑑₁𝑏} and {𝑐𝑎𝑎,𝑎𝑑₁,𝑑₁𝑏} are not comma-free codes as the concatenations 𝑎𝑐𝑎.𝑑₁𝑏 and 𝑐𝑎𝑎.𝑑₁𝑏 lead to 𝑎𝑑₁ in contradiction with Definition 2.2. With the conjugation class {𝑏𝑏𝑐,𝑏𝑐𝑏,𝑐𝑏𝑏}, only 𝑐𝑏𝑏 can belong to 𝑋. Indeed, {𝑏𝑏𝑐,𝑎𝑑₁,𝑑₁𝑏} and {𝑏𝑐𝑏,𝑎𝑑₁,𝑑₁𝑏} are not comma-free codes as the concatenations 𝑎𝑑₁.𝑏𝑏𝑐 and 𝑎𝑑₁.𝑏𝑐𝑏 lead to 𝑑₁𝑏 in contradiction with Definition 2.2. Similarly, 𝑎𝑎𝑑 and 𝑑𝑏𝑏 must belong to 𝑋. Moreover, with the conjugation class {𝑎𝑐𝑏,𝑐𝑏𝑎,𝑏𝑎𝑐}, only 𝑎𝑐𝑏 can belong to 𝑋.
Now, we have: (i)𝑎𝑐𝑑∉𝑋 (otherwise {𝑎𝑎𝑐,𝑎𝑐𝑑,𝑑𝑏𝑏} is not a comma-free code);(ii)𝑐𝑑𝑎∉𝑋 (otherwise {𝑐𝑑𝑎,𝑎𝑐𝑏,𝑐𝑏𝑏} is not a comma-free code); (iii)𝑑𝑎𝑐∉𝑋 (otherwise {𝑎𝑎𝑑,𝑑𝑎𝑐,𝑎𝑐𝑏} is not a comma-free code). So, no element in the conjugation class {𝑎𝑐𝑑,𝑐𝑑𝑎,𝑑𝑎𝑐} belongs to 𝑋. This is a contradiction.

4. Extended Hierarchy

The previous hierarchy of trinucleotide circular codes [23] is now extended with these new 𝐷𝐿𝐷 and 𝐿𝐷𝐿 codes. By Proposition 3.10, the set of 𝐷𝐿𝐷-strong 20-trinucleotide circular codes is empty. Moreover, by Proposition 3.11, the set of 𝐿𝐷𝐿-strong 20-trinucleotide circular codes coincide with the set of trinucleotide comma-free codes (set 𝐼²). With the notations 𝐼^𝑛 and 𝑈^𝑛 (Notation 1), the hierarchy of the above recalled Proposition 2.17 is extended with these new strong trinucleotide circular codes as follows.

Proposition 4.1. With the 20-trinucleotide circular codes, the following chains of inclusions and equalities hold: ∅=𝐿𝐷𝐿∩𝐷𝐿𝐷⊂𝐿𝐷𝐿∪𝐷𝐿𝐷=𝐿𝐷𝐿=𝐼²⊂𝑈²=𝐼³𝐶⊂𝑈³𝐶=𝐼³⊂𝑈³=𝐼⁴𝐶⊂𝑈⁴𝐶=𝐼⁴⊂𝑈⁴=𝐼⁵𝐶⊂𝑈⁵𝐶=𝐼⁵=𝑈⁵. (4.1)

5. Coding of Trinucleotide Circular Codes with the Symmetric Group Σ_𝑛

We use the symmetric group Σ_𝑛 (e.g., [24]) to develop a coding of trinucleotide circular codes.

A permutation of a set 𝑋 is a bijection 𝜎 from 𝑋 into itself. Given a positive integer 𝑛, [𝑛] denotes the set {0,1,…,𝑛−1}. As [𝑛] has a natural total order 0<1<⋯<𝑛−1, a permutation 𝜎 of [𝑛] is the word 𝜎(0)𝜎(1)⋯𝜎(𝑛−1) giving the successive images of the elements of [𝑛]. Analogously, {𝑎_[𝑛]} denotes any totally ordered set {𝑎₀,𝑎₁,…,𝑎_𝑛−1}, 𝑎₀<𝑎₁<⋯<𝑎_𝑛−1, of 𝑛 elements. Also as a consequence of the total order, a permutation 𝜎 of {𝑎_[𝑛]} is the word 𝑎_𝜎(0)𝑎_𝜎(1)…𝑎_{𝜎(𝑛−1)} and by abuse of language, 𝜎 can also be considered as a permutation of [𝑛]. The symmetric group Σ_𝑛 denotes all the permutations of {𝑎_[𝑛]}.

Recall that |𝑋| denotes the number of elements of a set 𝑋. Recall that if 𝑤=𝑤(0)𝑤(1)⋯𝑤(𝑘−1) is a word of length 𝑘 on the alphabet 𝒜, then Alph(𝑤)={𝑤(0),𝑤(1),…,𝑤(𝑘−1)}. So, Alph(𝑤) is the set of the letters of 𝒜 having at least one occurrence in 𝑤.

A permutation of {𝑎_[𝑛]} can be represented by a word of length 𝑛−1. Clearly, the prefix of length 𝑛−1 of the word 𝑎_𝜎(0)…𝑎_{𝜎(𝑛−1)} uniquely determines 𝜎. There are also four other cases to represent the elements of Σ_𝑛 by words of length 𝑛−1: 𝑖<𝑗 and 𝜎(𝑖)<𝜎(𝑗); 𝑖<𝑗 and 𝜎(𝑖)>𝜎(𝑗); 𝑖>𝑗 and 𝜎(𝑖)<𝜎(𝑗); 𝑖>𝑗 and 𝜎(𝑖)>𝜎(𝑗). We begin with the case 𝑖<𝑗 and 𝜎(𝑖)>𝜎(𝑗).

For a given ℎ∈[𝑛−1], {𝑎_[ℎ]} denotes the subset of [𝑛−1] containing its first ℎ elements 𝑎₀,…,𝑎_ℎ−1. For a given 𝑖∈[𝑛] and for a permutation 𝜎 of {𝑎_[𝑛]}, the set 𝑟^𝜎_𝑖 is defined as follows: 𝑟^𝜎_𝑖={𝑎_[𝜎(𝑖)]}∩Alph(𝑎_𝜎(𝑖+1)⋯𝑎_{𝜎(𝑛−1)}) contains the elements of {𝑎_[𝜎(𝑖)]}={𝑎₀,…,𝑎_{𝜎(𝑖)−1}} having one occurrence in 𝑎_𝜎(𝑖+1)⋯𝑎_{𝜎(𝑛−1)}, the suffix of length 𝑛−𝑖−1 of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}. Consequently, |𝑟^𝜎_𝑖| counts the number of elements 𝑗 of [𝑛] such that 𝑖<𝑗 and 𝑎_𝜎(𝑖)>𝑎_𝜎(𝑗). In other words, |𝑟^𝜎_𝑖| counts the number of elements 𝑎_𝑘 of {𝑎_[𝑛]} such that 𝑎_𝑘<𝑎_𝜎(𝑖) and 𝑎_𝑘 is on the right of 𝑎_𝜎(𝑖) in the word 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}. Put 𝑟(𝑖)=|𝑟^𝜎_𝑖| and let the code of 𝜎 be the word 𝑟(0)𝑟(1)⋯𝑟(𝑛−1) denoted by 𝑟(𝜎).

For a given permutation 𝜎, 𝑟(0) is the number of the letters of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)} that are strictly smaller than 𝑎_𝜎(0) or equivalently, the number of the elements of the alphabet {𝑎_[𝑛]} that are strictly smaller than the leftmost letter 𝑎_𝜎(0), and by the choice of the alphabet, this number is exactly 𝜎(0) and belongs to [𝑛−0]=[𝑛]. Then, 𝑟(1) is the number of the letters of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)} that are strictly smaller than 𝑎_𝜎(1) and on the right of 𝑎_𝜎(1) or equivalently, the number of the elements of the alphabet {𝑎_[𝑛]}−{𝜎(0)} that are strictly smaller than 𝑎_𝜎(1) and this number belongs to [𝑛−1]. And so on until 𝑟(𝑛−2) which is the number of the letters of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)} that are strictly smaller than 𝑎_{𝜎(𝑛−2)} and on the right of 𝑎_{𝜎(𝑛−2)} or equivalently, the number of the elements of the two-letter alphabet {𝑎_[𝑛]}−{𝜎(0),…,𝜎(𝑛−3)}={𝜎(𝑛−2),𝜎(𝑛−1)} that are strictly smaller than 𝑎_{𝜎(𝑛−2)} and this number belongs to [𝑛−(𝑛−2)]=[2], that is, with only values 0 or 1. Finally, 𝑟(𝑛−1) is the number of the letters of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)} that are strictly smaller than 𝑎_{𝜎(𝑛−1)} and on the right of 𝑎_{𝜎(𝑛−1)} or equivalently, the number of the elements of the one-letter alphabet {𝑎_[𝑛]}−{𝜎(0),…,𝜎(𝑛−2)}={𝜎(𝑛−1)} that are strictly smaller than 𝑎_{𝜎(𝑛−1)} and this number belongs to [𝑛−(𝑛−1)]=[1], that is, with value equal to 0. Thus, 𝑟(0)∈[𝑛],𝑟(1)∈[𝑛−1],…,𝑟(𝑖)∈[𝑛−𝑖] and 𝑟(0)𝑟(1)⋯𝑟(𝑛−1) belongs to a set of cardinality 𝑛! which is exactly the cardinality of Σ_𝑛.

Clearly, if 𝜎 and 𝜏 are two different permutations of {𝑎_[𝑛]}, then 𝑟(𝜎)≠𝑟(𝜏). Indeed, let 𝑘 be the maximum integer such that 𝑎_𝜎(𝑘)=𝑎_𝜏(𝑘). Without loss of generality, suppose that 𝑎_𝜎(𝑘+1)<𝑎_𝜏(𝑘+1). As Alph(𝑎_𝜎(𝑘+1)⋯𝑎_{𝜎(𝑛−1)})=Alph(𝑎_𝜏(𝑘+1)⋯𝑎_{𝜏(𝑛−1)}), then |𝑟^𝜎_𝑘+1|<|𝑟^𝜏_𝑘+1|. So, 𝑟(𝜎) is different from 𝑟(𝜏).

Example 5.1. The code of the permutation 𝜎=𝑎₄𝑎₆𝑎₂𝑎₁𝑎₃𝑎₀𝑎₅ of {𝑎_[7]} is 𝑟(𝜎)=452110.

The correspondence 𝜌∶𝜎→𝑟(𝜎) is an injective map between two finite sets of same cardinality (𝑛!). So 𝜌 is a bijection and to each 𝑟(𝜎) corresponds a unique 𝜎. The following algorithm allows the permutation 𝜎 from the code 𝑟(𝜎) to be retrieved.

Algorithm 1 (principle). Initialisation 𝑎_𝜎(0)=𝑎_𝑟(0); only one element, say 𝑎_𝛼, in {𝑎_[𝑛]}−{𝑎_𝜎(0)} can verify 𝑟(1)=|{𝑎_𝜁∈{𝑎_[𝑛]}−{𝑎_𝜎(0)}∣𝑎_𝛼>𝑎_𝜁}|, so 𝑎_𝜎(1)=𝑎_𝛼; only one element, say 𝑎_𝛽, in {𝑎_[𝑛]}−{𝑎_𝜎(0),𝑎_𝜎(1)} can verify 𝑟(2)=|{𝑎_𝜁∈{𝑎_[𝑛]}−{𝑎_𝜎(0),𝑎_𝜎(1)}∣𝑎_𝛽>𝑎_𝜁}|, so 𝑎_𝜎(2)=𝑎_𝛽; repeat this procedure until all the elements {𝑎_𝜎(0),…,𝑎_{𝜎(𝑛−2)}} are found; finally 𝑎_{𝜎(𝑛−1)} is the unique value in {𝑎_[𝑛]}−{𝑎_𝜎(0),…,𝑎_{𝜎(𝑛−2)}}.

Remark 5.2. In general, 𝑟(𝑖+1)⋯𝑟(𝑛−1) is the code of the permutation 𝑎_𝜎(𝑖+1)⋯𝑎_{𝜎(𝑛−1)} on the totally ordered alphabet {𝑎_𝜎(𝑖+1),…,𝑎_{𝜎(𝑛−1)}}.

Example 5.3. Consider the previous example with the permutation 𝜎 of {𝑎_[7]} having the code 𝑟(𝜎)=452110. As 𝑟(0)=4, then 𝑎_𝜎(0)=𝑎₄; 𝑎_𝜎(1)=𝑎₆ as {𝑎₀,𝑎₁,𝑎₂,𝑎₃,𝑎₅,𝑎₆} contains 𝑟(1)=5 elements strictly smaller; 𝑎_𝜎(2)=𝑎₂ as {𝑎₀,𝑎₁,𝑎₂,𝑎₃,𝑎₅} contains 𝑟(2)=2 elements strictly smaller; 𝑎_𝜎(3)=𝑎₁ as {𝑎₀,𝑎₁,𝑎₃,𝑎₅} contains 𝑟(3)=1 element strictly smaller; 𝑎_𝜎(4)=𝑎₃ as {𝑎₀,𝑎₃,𝑎₅} contains 𝑟(4)=1 element strictly smaller; 𝑎_𝜎(5)=𝑎₀ as {𝑎₀,𝑎₅} contains 𝑟(5)=0 element strictly smaller; finally, 𝑎_𝜎(6)=𝑎₅ as {𝑎_[7]}−{𝑎_𝜎(0),𝑎_𝜎(1),𝑎_𝜎(2),𝑎_𝜎(3),𝑎_𝜎(4),𝑎_𝜎(5)}={𝑎_[7]}−{𝑎₄,𝑎₆,𝑎₂,𝑎₁,𝑎₃,𝑎₀}={𝑎₅}. So, the permutation 𝜎 is 𝑎₄𝑎₆𝑎₂𝑎₁𝑎₃𝑎₀𝑎₅.

For a given permutation 𝜎, we can also define the sets 𝑙^𝜎_𝑖={𝑎_[𝜎(𝑖)]}∩Alph(𝑎_𝜎(0)⋯𝑎_{𝜎(𝑖)−1}), 𝑅^𝜎_𝑖=({𝑎_[𝑛]}−{𝑎_{[𝜎(𝑖)+1]}})∩Alph(𝑎_𝜎(𝑖+1)⋯𝑎_{𝜎(𝑛−1)}) and 𝐿^𝜎_𝑖=({𝑎_[𝑛]}−{𝑎_{[𝜎(𝑖)+1]}})∩Alph(𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}). The set 𝑙^𝜎_𝑖 consists of the elements of {𝑎_[𝜎(𝑖)]}={𝑎_𝜎(0),…,𝑎_{𝜎(𝑖)−1}} that have one occurrence in the prefix of length 𝑖 of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}. Its cardinality |𝑙^𝜎_𝑖| counts the number of elements 𝑗 of [𝑛] such that 𝑗<𝑖 and 𝜎(𝑗)<𝜎(𝑖) or, in other words, |𝑙^𝜎_𝑖| counts the number of elements 𝑎_𝑘 of {𝑎_[𝑛]} such that 𝑎_𝑘<𝑎_𝜎(𝑖) and 𝑎_𝑘 is on the left of 𝑎_𝜎(𝑖) in 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}. Similarly, the set 𝑅^𝜎_𝑖 consists of the elements of {𝑎_[𝑛]}−{𝑎_{[𝜎(𝑖)+1]}}={𝑎_𝜎(𝑖)+1,…,𝑎_𝑛−1} that have one occurrence in 𝑎_𝜎(𝑖+1)⋯𝑎_{𝜎(𝑛−1)}, the suffix of length 𝑛−𝑖−1 of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}. Its cardinality |𝑅^𝜎_𝑖| counts the number of elements 𝑗 of [𝑛] such that 𝑗>𝑖 and 𝜎(𝑗)>𝜎(𝑖) or, in other words, |𝑅^𝜎_𝑖| counts the number of elements 𝑎_𝑘 of {𝑎_[𝑛]} such that 𝑎_𝑘>𝑎_𝜎(𝑖) and 𝑎_𝑘 is on the right of 𝑎_𝜎(𝑖) in 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}. Finally, the set 𝐿^𝜎_𝑖 consists of the elements of {𝑎_[𝑛]}−{𝑎_{[𝜎(𝑖)+1]}}={𝑎_𝜎(𝑖)+1,…,𝑎_𝑛−1} that have one occurrence in the prefix of length 𝑖 of 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}. Its cardinality |𝐿^𝜎_𝑖| counts the number of elements 𝑗 of [𝑛] such that 𝑗<𝑖 and 𝜎(𝑗)>𝜎(𝑖) or, in other words, |𝐿^𝜎_𝑖| counts the number of elements 𝑎_𝑘 of {𝑎_[𝑛]} such that 𝑎_𝑘>𝑎_𝜎(𝑖) and 𝑎_𝑘 is on the left of 𝑎_𝜎(𝑖) in 𝑎_𝜎(0)⋯𝑎_{𝜎(𝑛−1)}.

There are trivial relations𝑙^𝜎_𝑖+𝐿^𝜎_𝑖𝑟=𝑖,_𝜎𝑖+𝑅^𝜎_𝑖𝑟=𝑛−𝑖−1,_𝜎𝑖+𝑙^𝜎_𝑖(𝑖),𝑅=𝜎_𝜎𝑖+𝐿^𝜎_𝑖=𝑛−𝜎(𝑖)−1. (5.1)

For a given permutation, 𝑙^𝜎_𝑖, 𝑅^𝜎_𝑖, and 𝐿^𝜎_𝑖 allow the construction of three other codes, namely, 𝑙(0)𝑙(1)⋯𝑙(𝑛−1), 𝑅(0)𝑅(1)⋯𝑅(𝑛−1) and 𝐿(0)𝐿(1)⋯𝐿(𝑛−1), which have similar properties to the code 𝑟(0)𝑟(1)⋯𝑟(𝑛−1). These relations can retrieve more efficiently the permutation 𝜎 from the code 𝑟(𝜎). For the interesting case 𝑛=4 of this paper, an efficient algorithm is given.

Algorithm 2 (principle). Initialisation 𝑎_𝜎(0)=𝑎_𝑟(0); Consider {𝜎(1),𝜎(2),𝜎(3)} and let {𝜎(1),𝜎(2),𝜎(3)}={𝛼,𝛽,𝛾} with 𝛼<𝛽<𝛾.
If 𝑟(1)=2, then 𝑎_𝜎(1)=𝑎_𝛾 and, if 𝑟(2)=1, then 𝑎_𝜎(2)𝑎_𝜎(3)=𝑎_𝛽𝑎_𝛼 or, if 𝑟(2)=0, then 𝑎_𝜎(2)𝑎_𝜎(3)=𝑎_𝛼𝑎_𝛽.
If 𝑟(1)=1, then 𝑎_𝜎(1)=𝑎_𝛽 and, if 𝑟(2)=1, then 𝑎_𝜎(2)𝑎_𝜎(3)=𝑎_𝛾𝑎_𝛼 or, if 𝑟(2)=0, then 𝑎_𝜎(2)𝑎_𝜎(3)=𝑎_𝛼𝑎_𝛾.
If 𝑟(1)=0, then 𝑎_𝜎(1)=𝑎_𝛼 and, if 𝑟(2)=1, then 𝑎_𝜎(2)𝑎_𝜎(3)=𝑎_𝛾𝑎_𝛽 or, if 𝑟(2)=0, then 𝑎_𝜎(2)𝑎_𝜎(3)=𝑎_𝛽𝑎_𝛾.
The number 𝑟(1)𝑟(2) is the code of the permutation 𝑎_𝜎(𝛼)𝑎_𝜎(𝛽)𝑎_𝜎(𝛾) on {𝑎_𝛼,𝑎_𝛽,𝑎_𝛾}.

Example 5.4. Consider the permutation 𝜎 of {𝑎_[4]} having 111 as its code. Clearly, 𝑎_𝜎(0)=𝑎₁. Then, the considered set {𝜎(1),𝜎(2),𝜎(3)}={𝛼,𝛽,𝛾} is {𝑎₀,𝑎₂,𝑎₃}. As 𝑟(1)=1, then 𝑎_𝜎(1)=𝑎_𝛽=𝑎₂ and as 𝑟(2)=1, then 𝑎_𝜎(2)𝑎_𝜎(3)=𝑎_𝛾𝑎_𝛼=𝑎₃𝑎₀. So, the permutation 𝜎 is 𝑎₁𝑎₂𝑎₃𝑎₀.

Finally, the code of a permutation 𝜎(𝐴)𝜎(𝐶)𝜎(𝐺)𝜎(𝑇) on the genetic alphabet 𝒜₄ (𝐴<𝐶<𝐺<𝑇) can easily be computed by putting 𝐴=𝑎₀, 𝐶=𝑎₁, 𝐺=𝑎₂ and 𝑇=𝑎₃. Similarly, for the totally ordered alphabet {𝑎,𝑏,𝑐,𝑑} (𝑎<𝑏<𝑐<𝑑) in Section 5, the code of a permutation is obtained by putting 𝑎=𝑎₀, 𝑏=𝑎₁, 𝑐=𝑎₂ and 𝑑=𝑎₃.

6. Role of the Symmetric Group Σ₄

We put 𝑎=𝐴, 𝑏=𝐶, 𝑐=𝐺 and 𝑑=𝑇 and identify the elements of the symmetric group Σ₄ over {𝑎,𝑏,𝑐,𝑑} (𝑎<𝑏<𝑐<𝑑) with the 24 permutations of the word 𝑎𝑏𝑐𝑑. We denote the permutations by their codes (Table 1(c)).

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}.

We wish to point out that a computer calculus confirms that the 20-trinucleotide comma-free codes are exactly the 𝐿𝐷𝐿-strong 20-trinucleotide circular codes. These codes are partitioned into 28 classes: 𝐶₁,𝐶₂,…,𝐶₂₈. There are four classes containing six codes each (Table 2(a)), 16 classes containing 12 codes each (Table 2(b)), and eight classes containing 24 codes each (Table 2(c)). For each class, we give explicitely the list (in lexicographical order) of trinucleotides: the first (in lexicographical order) 𝐿𝐷𝐿-strong 20-trinucleotide circular code 𝑋 (pattern of the class) and the codes of the permutations of Σ₄ (Table 1(c)) on 𝑋 giving the other 𝐿𝐷𝐿-strong 20-trinucleotide circular codes of the class. The classes are lexicographically ordered according to the patterns of classes.

Table 2 

(a) The four classes having each six 𝐿𝐷𝐿-strong 20-trinucleotide circular codes. Each class is described by its pattern and the five codes of the permutations of the symmetric group Σ₄ on the pattern allow the other five 𝐿𝐷𝐿-strong 20-trinucleotide circular codes of the class to be deduced. (b) The 16 classes having each 12 𝐿𝐷𝐿-strong 20-trinucleotide circular codes. Each class is described by its pattern and the 11 codes of the permutations of the symmetric group Σ₄ on the pattern allow the other 11 𝐿𝐷𝐿-strong 20-trinucleotide circular codes of the class to be deduced. (c) The eight classes having each 24 𝐿𝐷𝐿-strong 20-trinucleotide circular codes. In this case, each class is only described by its pattern as the other 23 𝐿𝐷𝐿-strong 20-trinucleotide circular codes are obtained with the 23 permutations of the symmetric group Σ₄.

Table 3 

The prefixes and suffixes of the pattern of the 28 classes of 𝐿𝐷𝐿-strong 20-trinucleotide circular codes (Tables 2(a)–2(c)). For each pattern 𝑋, the subset 𝐿₁ (resp. 𝐿₃) of {𝑎,𝑏,𝑐,𝑑} consists of the letters 𝑙₁ (resp. 𝑙₃) that appear at least once in prefix (resp. suffix) position of the trinucleotides of 𝑋. Similarly, the subset 𝐷₁ (resp. 𝐷₂) of {𝑎,𝑏,𝑐,𝑑}² consists of the diletters 𝑑₁ (resp. 𝑑₂) that appear at least once in prefix (resp. suffix) position of the trinucleotides of 𝑋.

Moreover, a computer calculus describes the properties of prefixes and suffixes for the 28 classes of 𝐿𝐷𝐿-strong 20-trinucleotide circular codes 𝑋. The set 𝐿₁ is formed by the letters 𝑙₁ in the first position of the trinucleotides of 𝑋 and the set 𝐿₃, by the letters 𝑙₃ in the last position of the trinucleotides of 𝑋. The set 𝐷₁ is formed by the diletters 𝑑₁ in prefix position of the trinucleotides of 𝑋 and the set 𝐷₂, by the diletters 𝑑₂ in suffix position of the trinucleotides of 𝑋. Eight classes have both four letters in 𝐿₁ and 𝐿₂ (𝐶₁–𝐶₄, 𝐶₈, 𝐶₁₀, 𝐶₁₇, 𝐶₂₀). Ten classes have four letters in 𝐿₁ and three letters in 𝐿₂ (𝐶₅–𝐶₇, 𝐶₉, 𝐶₁₁, 𝐶₁₂, 𝐶₂₁–𝐶₂₄). Reciprocally, ten classes have four letters in 𝐿₂ and three letters in 𝐿₁ (𝐶₁₃–𝐶₁₆, 𝐶₁₈, 𝐶₁₉, 𝐶₂₅–𝐶₂₈). Three classes have nine diletters in 𝐷₁ (𝐶₅, 𝐶₇, 𝐶₂₁) and similarly, three classes have nine diletters in 𝐷₂ (𝐶₁₆, 𝐶₁₈, 𝐶₂₈). Only the class 𝐶₂₈ has six diletters in 𝐷₁ and nine diletters in 𝐷₂ and similarly, only the class 𝐶₂₁ has six diletters in 𝐷₂ and nine diletters in 𝐷₁. All the sets 𝐷₁∩𝐷₂ are obviously empty.

These tables and the other symmetric relations identified before (e.g., Proposition 6 of [23]) suggest that the symmetric group Σ₄ can have a very important role in the study of these trinucleotide circular codes.

Acknowledgments

We thank Jacques Justin for his advices. The second author thanks the Dipartimento di matematica “U. Dini” for giving him a friendly hospitality.