Proposition 3. The class of self-complementary circular codes with both and in the class of circular codes is non-empty.

Proof. Consider, for example, the following set of 20 trinucleotides

It is enough to prove that is a self-complementary circular code and that its two conjugated classes and are also circular codes.

is a self-complementary circular code.

is self-complementary. Obvious by inspection.

is a circular code. We use Proposition 1 [17]. By way of contradiction, suppose that admits a 5LDCN. As can be, , or, it is enough to prove that each choice leads to a contradiction.

1) If then there is no possible as is not a suffix of any trinucleotide of, contradiction.

2) If, there are three possible:

• if (a) or (b) then (c) but there is no possible as is not a prefix of any trinucleotide of, contradiction

• if (d), there is a contradiction as no trinucleotide of has a prefix.

3) If, there are six possible:

• if or, contradiction (a) and (b)

• if then, contradiction (c)

• if or then or:

if, there are three possible: if or then, similarly to (c), contradiction, and if, similarly to (d), contradiction

if, contradiction (c)

• if, contradiction (d).

4) If, similarly to (c), contradiction.

As, for each letter, we cannot complete the assumed 5LDCN for, we are in contradiction. Hence, is a circular code.

is a circular code. We have to prove that

is a circular code. By way of contradiction, assume that admits a 5LDCN.

1) If, there are four possible:, , and, but no possible, contradiction.

2) If, there are three possible:, and, but no possible, contradiction.

3) If, there are six possible:, and, and the cases, and already seen, but no possible, contradiction.

4) If, there is no possible, contradiction.

Hence, is also a circular code.

is a circular code. Finally, we have to prove that

is a circular code. By way of contradiction, assume that admits a 5LDCN.

1) If, there is no possible, contradiction.

2) If, there are six possible:, , , , and, but no possible, contradiction.

3) If, there are three possible:, and which are cases already seen, contradiction.

4) If, there are four possible:, , and, but no possible, contradiction.

Hence, as and, is also a circular code.

Proposition 4. The class of self-complementary circular codes having 20 elements with neither nor in the class of circular codes is non-empty.

Proof. Consider, for example, the following set of 20 trinucleotides

It is enough to prove that is a self-complementary circular code and that neither its conjugated class nor its conjugated class are circular codes.

is a self-complementary circular code.

is self-complementary. Obvious by inspection.

is a circular code. We use Proposition 1 [17]. By way of contradiction, assume that admits a 5LDCN.

1) If then there is one possible but no possible, contradiction.

2) If, there are two possible:

• if then (a) and (b) but there is no possible, contradiction

• if (c) then there is no possible, contradiction.

3) If we have seven possible:

• if then or:

if (d) then or:

- if then and but there is no possible, contradiction

- if then there is no possible, contradiction

if, contradiction (a)

• if, similarly to (b), contradiction

• if, or then, contradiction (a)

• if then or, contradiction (a) and (d)

• if, contradiction (c).

4) If, similarly to (a), contradiction.

Hence, is a circular code.

is not a circular code. We have

We use a technique developed in [23]. Observe that contains So,

is a 5LDCN for this 4-element subset of and, a fortiori, for itself which, consequently, is not a circular code.

is not a circular code. We have

We again use a technique developed in [23]. Remark that contains. So,

is a 5LDCN for this 4-element subset of and, a fortiori, for itself which, consequently, is not a circular code.

We need the propositions hereafter and, in particular the following one which states a general property of the involutional antiisomorphisms such as the complementary map.

Proposition 5. A subset of is a circular code if and only if is a circular code.

Proof. Suppose, first, that is not a circular code and that is a circular code. So has a 5LDCN. This means that there are 13 nucleotides, say

such that the trinucleotides

and

Now, consider the sequence

All the following trinucleotides belong to:

and

as they are the complement of trinucleotides in. So, admits a 5LDCN and it cannot be a circular code. Contradiction.

The case is a circular code and is not a circular code is similar.

Proposition 6. Let be a self-complementary subset of. If is partitioned into three classes such that two of them are the complement of each other then necessarily the third one is self-complementary.

Proof. Let, and be the three classes of an arbitrary partition of and suppose that and are complementary, i.e. and satisfy. Let be a trinucleotide of. We claim that. Indeed, in the opposite case, should not be the complement of because. We also claim that. Indeed, in the opposite case, should not be the complement of because. It remains the case. So, is self-complementary.

Remark 1. Clearly, if, and constitute an arbitrary partition of then the self-complementarity of is not enough to ensure that and are complementary of each other. This remark is again true if, in addition, is a self-complementary circular code having 20 elements. Indeed in this case, it is easy to make a partition in two classes and that are not complementary of each other. Any case, if we consider the partition of in the three classes given by a self-complementary trinucleotide circular code having 20 elements and by its two conjugated classes and then the necessary and sufficient condition holds (Proposition 7 below).

Proposition 7. A trinucleotide circular code having 20 elements is self-complementary if and only if and are complement of each other.

Proof if part. It is a trivial consequence of Proposition 6.

Only if part. Suppose that is self-complementary and consider the partition, and of. Suppose that the trinucleotide, say, belongs to. Then, also

.

We have

and

.

As is a generic trinucleotide of and as

and

then is the complement of.

As a consequence, we have the following proposition.

Proposition 8. If a trinucleotide circular code having 20 elements is self-complementary then either 1) and are both circular codes or 2) and are not circular codes (both have a necklace).

Proof. We have four possibilities:

is a circular code and is a circular code;

is a circular code and is not a circular code;

is not a circular code and is a circular code;

is not a circular code and is not a circular code.

Now, by applying Propositions 3 and 4, we have that the first and the last possibilities can be effectively realized.

Suppose that, by way of contradiction, the second possibility is realized. So, is a circular code. By Proposition 7, we have. So, by Proposition 5, must also be a circular code. Contradiction.

Suppose that, by way of contradiction, the third possibility is realized. So, is a circular code. By Proposition 7, we have. So, by Proposition 5, must also be a circular code. Contradiction.

So, only the first and the last possibilities can occur.

Hence, our proposition holds.

Proposition 9. The 528 self-complementary circular codes having 20 elements are partitioned into two classes: one class contains codes with the two permuted sets and which are both circular codes while the other class contains codes with the two permuted sets and which both are not circular codes.

Proof. It is enough to apply Proposition 8 to each of the 528 trinucleotide circular codes having 20 elements.

4. Acknowledgements

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

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