
Sequence embedding?
Introduction:
I'm an Electronics Engineer with a love for math so due to my background pardon me if my description is inadequate (I tried hard to make it formally correct).
Context and motivation:
I have been working with Sidel'nikov sequences and through exploratory analysis I found that some sequences are contained in lengthier ones.
Conditions:
Consider only binary sequences and for a Fermat prime.
Although not the original definition, the positions of "1" in a Sidel'nikov sequence can be determined using the Zech log,
or alternatively with a primitive element of and . To be entirely correct belongs to a prime residue group, so but since we are restricting ourselves to Fermat primes then we can just consider to be in the set of all the odd integers less than .
A numerical example using Mathematica:
For a sequence of length 16 with primitive element 3,
Sort[Table[Zech[n,3,17],{n,1,171,2]]
{4, 6, 7, 10, 11, 12, 13, 15}
For a sequence of length 256 with primitive element 3, and taking values (mod 16)
Mod[Sort[Table[Zech[n,3,257],{n,1,2571,2]],16]
{4,6,7,9,13,...,5,10,11,12,13,15}
where we can see that the lengthier sequence is wrapped by the smaller seq.
I'm interested in the values of that satisfy both congruences. So the question is: what values of satisfy simultaneously
and ?
I've tried to attack the problem using instead
but one of the problems is that of ordering. Any advise please?