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,1,17-1,2]]

{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,1,257-1,2]],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?