# Operating on a set of sequences - such as adding sequences and so on possible even when sequence is coded as number?

Suppose that there is a way to code some set of sequences into number. Then one is given one number. Suppose that we want to multiply all numbers in each sequence by that number and get the coded number of the resulting set of sequences. But we want to do this without decoding the number that uniquely represents soem set of sequences into sequences and operating on them.

Is there any way of coding that satisfies the above condition? All sequences are assumed to have same cardinality.

All sequences are of finite cardinality, all numbers are integers.

Additional question: And with all conditions above, would it be possible to append some number at the end of all sequences and get the coded number of the resulting set without decoding the sequences fully?

A set of sequences here is a set of $k$-tuples and $k$ can be set whatever one wishes to. But each set that is given as an input is assumed to have tuples that are of the same cardinality. And any number, whether it is a number that is to be multiplied, appended or is in a tuple, can be chosen from any number in integer range.

By "some set of sequences", I mean: {{elements of sequence 1}, {elements of sequence 2}, ..}. This set is of finite cardinality - that is, it contatins only finite sequences. And each sequence also contains finite elements. There are therefore countably infinite number of possible "some set of sequences".

The coding method is set to be uniform (that is, the coding works for all $k$) or the following: let's say that to some set of sequences, we want to add more sequences into the set. Then it must be possible that without decoding, two sets can be union-ed and converted into the coded number.

I do not know whether such coding exists, and that's why I asked this question.

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Would you be satisfied to code a sequence $s=a,b,c,d,e$ as $n_s=2^a3^b5^c7^d11^e$ Then if you had a set $S=\lbrace t,u,v,w,x,y,z\rbrace$ of sequences in yous favorite order or in order of increasing $n$ then encode the set as $N_S=2^{n_t}3^{n_u}\cdots 13^{n_y}17^{n_z}.$ That doesn't meet all your requirements but it is a start. – Aaron Meyerowitz Mar 31 '13 at 7:38
Sounds like you're interested in something like homomorphic encryption minus the security aspect - en.wikipedia.org/wiki/Homomorphic_encryption – François G. Dorais Mar 31 '13 at 13:11