# Complete Quantum Information in the DNA Genetic Code

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## Abstract

**:**

## 1. Introduction

La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i qualimezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.(Galileo Galilei (1564–1642), Il Saggiatore, cap. 6).

## 2. DNA and the Genetic Code

#### 2.1. Main Properties of DNA and the Genetic Code

- Met, Trp: 2 singlets;
- Asn, Asp, Cys, Gln, Glu, His, Lys, Phe, Tyr: 9 doublets;
- Ile, Term: 2 triplets;
- Ala, Gly, Pro, Thr, Val: 5 quadruplets;
- Arg, Leu, Ser: 3 sextets.

#### 2.2. Theories about the Degeneracies within the Genetic Code

## 3. Symmetries and Quantum Information from the Characters of a Finite Group

#### 3.1. Minimal Informationally Complete Quantum Measurements

#### 3.2. The Character Table of a Finite Group and Quantum Information

## 4. Symmetries and Quantum Information in the Genetic Code

## 5. Symmetries of the Genetic Code, the Golden Ratio, and a Hyperelliptic Curve

#### 5.1. Points of $\mathcal{C}$ over Cyclotomic Fields

#### 5.2. The Group Law on the Jacobian J of the Hyperelliptic Curve $\mathcal{C}$

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The double-helical structure of DNA and base pairing. Complementary bases are held together as a pair by hydrogen bonds. Two hydrogen bonds connect T to A and three hydrogen bonds connect G to C. The image is borrowed from Wikipedia [36].

**Figure 2.**(

**a**) The generic codon table for RNA and (

**b**) the codon table colored by amino acid polar requirement values [12].

**Table 1.**The character table for the symmetric group $G={S}_{4}$. There are five conjugacy classes of elements of dimensions 1, 1, 2, 3, and 3, respectively. The size and the order of an element in the class is as shown. The last five rows of the table are the irreducible characters ${\kappa}_{i}$, $i=1,\dots ,5$, of G. To each of them is associated the rank of the Gram matrix defined in Section 3.1.

Class | 1 | 2 | 3 | 4 | 5 | Class |

Size | 1 | 3 | 6 | 8 | 6 | Size |

Order | 1 | 2 | 2 | 3 | 4 | Order |

Char | dim | Gram | ||||

${\kappa}_{1}$ | 1 | 1 | 1 | 1 | 1 | 5 |

${\kappa}_{2}$ | 1 | 1 | −1 | 1 | − 1 | 21 |

${\kappa}_{3}$ | 2 | 2 | 0 | −1 | 0 | ${d}^{2}$ |

${\kappa}_{4}$ | 3 | −1 | −1 | 0 | 1 | ${d}^{2}$ |

${\kappa}_{5}$ | 3 | −1 | 1 | 0 | −1 | ${d}^{2}$ |

**Table 2.**The character table for the symmetric group $G=2O=(48,28)$. There are eight conjugacy classes of elements of dimensions $1(\times 2)$, $2(\times 3)$, $3(\times 2)$, and 4. The size and the order of an element in the class is as shown. The last five rows of the table are the irreducible characters ${\kappa}_{i}$, $i=1,\cdots ,8$, of G. To each of them is associated the rank of the Gram matrix defined in Section 3.1.

Class | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Class |

Size | 1 | 1 | 8 | 6 | 12 | 8 | 6 | 6 | Size |

Order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | Order |

Char | dim | Gram | |||||||

${\kappa}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

${\kappa}_{2}$ | 1 | 1 | 1 | 1 | − 1 | 1 | −1 | −1 | 45 |

${\kappa}_{3}$ | 2 | 2 | −1 | 2 | 0 | −1 | 0 | 0 | 56 |

${\kappa}_{4}$ | 2 | −2 | −1 | 0 | 0 | 1 | $\sqrt{2}$ | $-\sqrt{2}$ | 59 |

${\kappa}_{5}$ | 2 | −2 | −1 | 0 | 0 | 1 | $-\sqrt{2}$ | $\sqrt{2}$ | 59 |

${\kappa}_{6}$ | 3 | 3 | 0 | −1 | 1 | 0 | −1 | −1 | 59 |

${\kappa}_{7}$ | 3 | 3 | 0 | −1 | −1 | 0 | 1 | 1 | 59 |

${\kappa}_{8}$ | 4 | −4 | 1 | 0 | 0 | −1 | 0 | 0 | 59 |

**Table 3.**For the group $G=(240,105)\cong {\mathbb{Z}}_{5}\u22ca2O$, the table provides the dimension of the representation, the rank of the Gram matrix obtained under the action of the 22 -dimensional Pauli group, the order of a group element in the class and a good assignment to an amino acid according to its polar requirement value. Bold characters are for faithful representations. There is an ‘exception’ for the assignment of the sextet ‘Leu’ that is assumed to occupy two 4-dimensional slots. All characters are informationally complete except for the ones assigned to ‘Stop’, ‘Leu’, ‘Pyl’, and ‘Sec’.

(240,105) | dimension | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

${\mathbb{Z}}_{5}\u22ca({\mathbb{Z}}_{2}.{S}_{4})$ | d-dit, d = 22 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ |

$\cong {\mathbb{Z}}_{5}\u22ca2O$ | amino acid | Met | Trp | Cys | Phe | Tyr | His | Gln | Asn | Lys | Glu | Asp |

order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | |

polar req. | 5.3 | 5.2 | 4.8 | 5.0 | 5.4 | 8.4 | 8.6 | 10.0 | 10.1 | 12.5 | 13.0 | |

(240,105) | dimension | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 |

d-dit, d = 22 | ${d}^{2}$ | 475 | 483 | 480 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

amino acid | Ile | Stop | Leu,Pyl,Sec | Leu | Val | Pro | Thr | Ala | Gly | Ser | Arg | |

order | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | |

polar req. | 4.9 | 4.9 | 5.6 | 6.6 | 6.6 | 7.0 | 7.9 | 7.5 | 9.1 |

**Table 4.**Excerpt of Table 3 with the assignment of codons to amino acids according to the standard genetic code. There is no ambiguity in such assignments except for codons UAG and UGA that code for ‘Stop’, as well as for ‘Pyl’ and ‘Sec’, respectively. According to our approach, this ambiguity is reflected in the characters being non informationally complete for the corresponding two slots. Bold characters are for faithful representations.

(240,105) | dimension | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

$\cong {\mathbb{Z}}_{5}\u22ca2O$ | amino acid | Met | Trp | Cys | Phe | Tyr | His | Gln | Asn | Lys | Glue | Asp |

bases | AUG | UGG | UGU | UUU | UAU | CAU | CAA | AAU | AAA | GAA | GAU | |

UGC | UUC | UAC | CAC | CAG | AAC | AAG | GAG | GAC | ||||

(240,105) | dimension | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 |

amino acid | Ile | Stop | Leu | Leu | Val | Pro | Thr | Ala | Gly | Ser | Arg | |

Pyl,Sec | ||||||||||||

bases | AUU | UAA | UUA | CUU | GUU | CCU | ACU | CGU | GGU | UCU | CGU | |

AUC | UAG | UUG | CUC | GUC | CCC | ACC | GCC | GGC | UCC | CGC | ||

AUA | UGA | UAG | CUA | GUA | CCA | ACA | GCA | GGA | UCA | GGA | ||

UGA | CUG | GUG | CCG | ACG | GCG | GGG | UCG | CGG | ||||

AGU | AGA | |||||||||||

AGC | AGG |

**Table 5.**The group $(240,106)\cong {\mathbb{Z}}_{5}\u22caGL(2,3)$ as another candidate for the assignments of amino acids. It has the same dimensions of representations and closely related group orders. One may postulate that the group $(240,106)$ encodes D-amino acids while the group $(240,105)$ encodes L-amino acids. For most naturally-occurring amino acids, the carbon alpha to the amino group has the L-configuration. Bold characters are for faithful representations.

(240,106) | dimension | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

$\cong {\mathbb{Z}}_{5}\u22caGL(2,3)$ | order | 1 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 8 | 8 | 10 |

d-dit, d = 22 | ${d}^{2}$ | ${d}^{2}$ | 483 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | |

(240,106) | dimension | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 |

order | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | |

d-dit, d = 22 | 482 | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | ${d}^{2}$ | 440 | 440 |

**Table 6.**The assignment of representations of groups $2O$ or $GL(2,3)$ to most essential amino acids in humans. There are two extra essential amino acids Val and Leu but they are not strictly essential from a metabolic perspective. Strictly indispensable amino acids are Trp, Lys, and Thr which cannot be submitted to transamination. Bold characters are for faithful representations.

dimension | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 4 | |

order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | |

(48,28) = 2O | d-dit, d = 8 | 8 | 45 | 56 | 59 | 59 | 59 | 59 | 59 |

(48,29) = GL(2,3) | d-dit, d = 8 | 56 | 58 | 59 | ${d}^{2}$ | ${d}^{2}$ | 63 | 63 | 61 |

amino acid | Met | Trp | Phe | His | Lys | Ile | Stop | Thr | |

polar req. | 5.3 | 5.2 | 5.0 | 8.4 | 10.1 | 4.9 | 6.6 |

**Table 7.**The character table for the group $G=(240,105)\cong {\mathbb{Z}}_{5}\u22ca2O$. The two right hand side columns are for the rank of the Gram matrix (with $d=22$) for the corresponding character and the assignment of an amino acid of the genetic code, respectively (see also Table 3 and Table 4 for further details). The notation in the entries is as follows: ${z}_{1}=-(\sqrt{5}+1)/2$, ${z}_{2}=\sqrt{5}-1$, ${z}_{3}=3(1+\sqrt{5})/2$, ${z}_{4}=\sqrt{2}$, the exponent * means the algebraic conjugation, e.g., ${z}_{1}^{*}=(\sqrt{5}-1)/2$, the ${r}_{i}$’s are the four real roots of the quartic curve ${x}^{4}-{x}^{3}-4{x}^{2}+4x+1=0$, i.e., ${r}_{1}\approx -1.956295$, ${r}_{2}\approx 1.338261$, ${r}_{3}\approx -0.209056$, and ${r}_{4}\approx 1.827091$.

${\kappa}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${\kappa}_{2}$ | 1 | 1 | 1 | 1 | −1 | 1 | 1 | 1 | −1 | −1 | 1 | 1 |

${\kappa}_{3}$ | 2 | 2 | −1 | 2 | 0 | 2 | 2 | −1 | 0 | 0 | 2 | 2 |

${\kappa}_{4}$ | 2 | 2 | 2 | 2 | 0 | ${z}_{1}$ | ${z}_{1}^{*}$ | 2 | 0 | 0 | ${z}_{1}$ | ${z}_{1}^{*}$ |

${\kappa}_{5}$ | 2 | 2 | 2 | 2 | 0 | ${z}_{1}^{*}$ | ${z}_{1}$ | 2 | 0 | 0 | ${z}_{1}^{*}$ | ${z}_{1}$ |

${\kappa}_{6}$ | 2 | −2 | −1 | 0 | 0 | 2 | 2 | 1 | ${z}_{4}$ | $-{z}_{4}$ | −2 | −2 |

${\kappa}_{7}$ | 2 | −2 | −1 | 0 | 0 | 2 | 2 | 1 | $-{z}_{4}$ | ${z}_{4}$ | −2 | −2 |

${\kappa}_{8}$ | 2 | 2 | −1 | 2 | 0 | ${z}_{1}$ | ${z}_{1}^{*}$ | −1 | 0 | 0 | ${z}_{1}$ | ${z}_{1}^{*}$ |

${\kappa}_{9}$ | 2 | 2 | −1 | 2 | 0 | ${z}_{1}^{*}$ | ${z}_{1}$ | −1 | 0 | 0 | ${z}_{1}^{*}$ | ${z}_{1}$ |

${\kappa}_{10}$ | 2 | 2 | −1 | 2 | 0 | ${z}_{1}$ | ${z}_{1}^{*}$ | −1 | 0 | 0 | ${z}_{1}$ | ${z}_{1}^{*}$ |

${\kappa}_{11}$ | 2 | 2 | −1 | 2 | 0 | ${z}_{1}^{*}$ | ${z}_{1}$ | −1 | 0 | 0 | ${z}_{1}^{*}$ | ${z}_{1}$ |

${\kappa}_{12}$ | 3 | 3 | 0 | −1 | −1 | 3 | 3 | 0 | 1 | 1 | 3 | 3 |

${\kappa}_{13}$ | 3 | 3 | 0 | −1 | 1 | 3 | 3 | 0 | −1 | −1 | 3 | 3 |

${\kappa}_{14}$ | 4 | −4 | 1 | 0 | 0 | 4 | 4 | −1 | 0 | 0 | −4 | −4 |

${\kappa}_{15}$ | 4 | −4 | −2 | 0 | 0 | ${z}_{2}$ | ${z}_{2}^{*}$ | 2 | 0 | 0 | $-{z}_{2}$ | $-{z}_{2}^{*}$ |

${\kappa}_{16}$ | 4 | −4 | −2 | 0 | 0 | ${z}_{2}^{*}$ | ${z}_{2}$ | 2 | 0 | 0 | $-{z}_{2}^{*}$ | $-{z}_{2}$ |

${\kappa}_{17}$ | 4 | −4 | 1 | 0 | 0 | ${z}_{2}$ | ${z}_{2}^{*}$ | −1 | 0 | 0 | $-{z}_{2}$ | $-{z}_{2}^{*}$ |

${\kappa}_{18}$ | 4 | −4 | 1 | 0 | 0 | ${z}_{2}$ | ${z}_{2}^{*}$ | −1 | 0 | 0 | $-{z}_{2}$ | $-{z}_{2}^{*}$ |

${\kappa}_{19}$ | 4 | −4 | 1 | 0 | 0 | ${z}_{2}^{*}$ | ${z}_{2}$ | −1 | 0 | 0 | $-{z}_{2}^{*}$ | $-{z}_{2}$ |

${\kappa}_{20}$ | 4 | −4 | 1 | 0 | 0 | ${z}_{2}^{*}$ | ${z}_{2}$ | −1 | 0 | 0 | $-{z}_{2}^{*}$ | $-{z}_{2}$ |

${\kappa}_{21}$ | 6 | 6 | 0 | −2 | 0 | ${z}_{3}$ | ${z}_{3}^{*}$ | 0 | 0 | 0 | ${z}_{3}$ | ${z}_{3}^{*}$ |

${\kappa}_{22}$ | 6 | 6 | 0 | −2 | 0 | ${z}_{3}^{*}$ | ${z}_{3}$ | 0 | 0 | 0 | ${z}_{3}^{*}$ | ${z}_{3}$ |

→ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ${d}^{2}$ | Met |

→ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ${d}^{2}$ | Trp |

→ | −1 | −1 | −1 | −1 | 2 | 2 | −1 | −1 | −1 | − 1 | ${d}^{2}$ | Cys |

→ | ${z}_{1}^{*}$ | ${z}_{1}^{*}$ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}^{*}$ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}^{*}$ | ${z}_{1}^{*}$ | ${d}^{2}$ | Phe |

→ | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}^{*}$ | ${z}_{1}^{*}$ | ${z}_{1}^{*}$ | ${z}_{1}$ | ${z}_{1}^{*}$ | ${z}_{1}^{*}$ | ${z}_{1}$ | ${z}_{1}$ | ${d}^{2}$ | Tyr |

→ | −1 | −1 | −1 | −1 | 0 | 0 | 1 | 1 | 1 | 1 | ${d}^{2}$ | His |

→ | −1 | −1 | −1 | −1 | 0 | 0 | 1 | 1 | 1 | 1 | ${d}^{2}$ | Gln |

→ | ${r}_{4}$ | ${r}_{3}$ | ${r}_{1}$ | ${r}_{2}$ | ${z}_{1}^{*}$ | ${z}_{1}$ | ${r}_{1}$ | ${r}_{2}$ | ${r}_{3}$ | ${r}_{4}$ | ${d}^{2}$ | Lys |

→ | ${r}_{2}$ | ${r}_{1}$ | ${r}_{4}$ | ${r}_{3}$ | ${z}_{1}$ | ${z}_{1}^{*}$ | ${r}_{4}$ | ${r}_{3}$ | ${r}_{1}$ | ${r}_{2}$ | ${d}^{2}$ | Glu |

→ | ${r}_{3}$ | ${r}_{4}$ | ${r}_{2}$ | ${r}_{1}$ | ${z}_{1}^{*}$ | ${z}_{1}$ | ${r}_{2}$ | ${r}_{1}$ | ${r}_{4}$ | ${r}_{3}$ | ${d}^{2}$ | Asp |

→ | 0 | 0 | 0 | 0 | −1 | −1 | 0 | 0 | 0 | 0 | ${d}^{2}$ | Ile |

→ | 0 | 0 | 0 | 0 | −1 | −1 | 0 | 0 | 0 | 0 | 475 | Stop |

→ | 1 | 1 | 1 | 1 | 0 | 0 | −1 | −1 | −1 | −1 | 483 | Leu, Pyl, Sec |

→ | $-{z}_{1}$ | $-{z}_{1}$ | $-{z}_{1}^{*}$ | $-{z}_{1}^{*}$ | 0 | 0 | ${z}_{1}^{*}$ | ${z}_{1}^{*}$ | ${z}_{1}$ | ${z}_{1}$ | 480 | Leu |

→ | $-{z}_{1}^{*}$ | $-{z}_{1}^{*}$ | $-{z}_{1}$ | $-{z}_{1}$ | 0 | 0 | ${z}_{1}$ | ${z}_{1}$ | ${z}_{1}^{*}$ | ${z}_{1}^{*}$ | ${d}^{2}$ | Val |

→ | $-{r}_{3}$ | $-{r}_{4}$ | $-{r}_{2}$ | $-{r}_{1}$ | 0 | 0 | ${r}_{2}$ | ${r}_{1}$ | ${r}_{4}$ | ${r}_{3}$ | ${d}^{2}$ | Pro |

→ | $-{r}_{4}$ | $-{r}_{3}$ | $-{r}_{1}$ | $-{r}_{2}$ | 0 | 0 | ${r}_{1}$ | ${r}_{2}$ | ${r}_{3}$ | ${r}_{4}$ | ${d}^{2}$ | Thr |

→ | $-{r}_{1}$ | $-{r}_{2}$ | $-{r}_{3}$ | $-{r}_{4}$ | 0 | 0 | ${r}_{3}$ | ${r}_{4}$ | ${r}_{2}$ | ${r}_{1}$ | ${d}^{2}$ | Ala |

→ | $-{r}_{2}$ | $-{r}_{1}$ | $-{r}_{4}$ | $-{r}_{3}$ | 0 | 0 | ${r}_{4}$ | ${r}_{3}$ | ${r}_{1}$ | ${r}_{2}$ | ${d}^{2}$ | Gly |

→ | 0 | 0 | 0 | 0 | $-{z}_{1}$ | $-{z}_{1}^{*}$ | 0 | 0 | 0 | 0 | ${d}^{2}$ | Ser |

→ | 0 | 0 | 0 | 0 | $-{z}_{1}^{*}$ | $-{z}_{1}$ | 0 | 0 | 0 | 0 | ${d}^{2}$ | Arg |

**Table 8.**The inverse of the 8 elements ${a}_{i}$ of bound 1 in the Jacobian J of the hyperelliptic curve $\mathcal{C}$ over $\mathbb{Q}\left({\zeta}_{5}\right)$. One gets $-{a}_{2}=(1,-{x}^{2}+x/2,2)$, $-{a}_{3}=(x,-{x}^{2}-1,2)$, $-{a}_{4}=(x,-{x}^{2}+1,2)$ and $-{a}_{5}=(x-1,-{x}^{2},2)$ of bounds 4, 5, 15, and 7, respectively.

points of J | $a1$ | $a2$ | $a3$ | $a4$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{7}$ | ${a}_{8}$ |

inverse | $a1$ | 4 | 5 | 15 | 7 | ${a}_{6}$ | ${a}_{7}$ | ? |

**Table 9.**The addition table between the 8 elements ${a}_{i}$ of bound 1 in the Jacobian J of the hyperelliptic curve $\mathcal{C}$ over $\mathbb{Q}\left({\zeta}_{5}\right)$. The 10 elements with bound 2 in the Jacobian are denoted ${b}_{i}$. For entries with bound not 1 or 2, only the bound is given. The points of the Jacobian with bounds 2 used in the table are ${b}_{1}=(x+2,-{x}^{2}+3,2)$, ${b}_{6}=(x-70/31,-{x}^{2}+221/31,2)$, ${b}_{8}=(x-6,-{x}^{2}+5,2)$, ${b}_{2}\approx (x+0.19821,-{x}^{2}+0.28285,2)$, ${b}_{9}\approx (x-1.30757,-{x}^{2}-2.47363,2)$, and ${b}_{10}\approx (x-2.00889,-{x}^{2}-1.01292,2)$.

sum | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{7}$ | ${a}_{8}$ |

${a}_{1}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{7}$ | ${a}_{8}$ |

${a}_{2}$ | . | 15 | 5 | 4 | 7 | ${a}_{5}$ | ${a}_{8}$ | ? |

${a}_{3}$ | . | . | ${b}_{6}$ | ${a}_{2}$ | 6 | ${b}_{8}$ | ${b}_{10}$ | 54 |

${a}_{4}$ | . | . | . | ${b}_{8}$ | ${b}_{6}$ | ${b}_{1}$ | ${b}_{2}$ | 14 |

${a}_{5}$ | . | . | . | . | 15 | ${a}_{2}$ | ${b}_{9}$ | ? |

${a}_{6}$ | . | . | . | . | . | ${a}_{1}$ | ${b}_{4}$ | ${b}_{9}$ |

${a}_{7}$ | . | . | . | . | . | . | ${a}_{1}$ | ${a}_{2}$ |

${a}_{8}$ | . | . | . | . | . | . | . | 15 |

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**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Complete Quantum Information in the DNA Genetic Code. *Symmetry* **2020**, *12*, 1993.
https://doi.org/10.3390/sym12121993

**AMA Style**

Planat M, Aschheim R, Amaral MM, Fang F, Irwin K. Complete Quantum Information in the DNA Genetic Code. *Symmetry*. 2020; 12(12):1993.
https://doi.org/10.3390/sym12121993

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, and Klee Irwin. 2020. "Complete Quantum Information in the DNA Genetic Code" *Symmetry* 12, no. 12: 1993.
https://doi.org/10.3390/sym12121993