[sdiy] Shift register sequence period

[Tim Stinchcombe]

Hi Dave,

> Maybe I'm misunderstanding your scenario, but when the
> register is all 
> zero, the feedback will be ~(0^0^0) = 1.  Here's the sequence 
> going thru 
> zero:
> 
>                    262137, 000000000000001111
>                    262138, 000000000000000111
>                    262139, 000000000000000011
>                    262140, 000000000000000001
>                    262141, 000000000000000000
>                    262142, 100000000000000000
>                    262143, 110000000000000000
>                    262144, 111000000000000000
>                    262145, 111100000000000000
>                    262146, 111110000000000000

What I was trying to describe was the situation at your step 262140, with
just the one '1' in it; as the feedback then is zero, the next step is the
all-zero case, 262141. If this were a *linear* FSR, this would put paid to
it, as if would then (necessarily for a linear FSR)feedback zero, and we'd
forever stay at zero. That we *do* get a non-zero feedback from the all-zero
state, and so the sequence keeps going, is what stops it being linear.

But actually there *is* another possibility: I think most of the theory is
devoted to linear, *homogeneous* recurring sequences, the 'homogeneous'
referring to the fact that there is no constant amount added to the feedback
(a '1' for binary case), 'constant' in that it is not derived from any of
the register stages; this is effectively what the inverter seems to be doing
in this circuit, but in either case, be it non-linear or inhomogeneous, it
is that which stops much of the theory applying, and hence the difficulty in
calcuating the sequence length directly (from some polynomial, assuming that
one could sensibly work it out!).

I'm still playing with the polynomials from the length-19 equivalent linear
FSR, but I'm having some difficulty with finding the 'order' of the big
nasty one in my last post. I haven't read enough of my book yet to decided
if I can find it easily, or if I'm going to need to do some brute force
polynomial division, which will either require finding a finite field
package for Mathematica, or a messy lot of programming that I don't think I
have the enthusiasm for.

Rest assured if I do get it sussed, I will let the list know (that's if
anyone else is still following this...!?).

Tim
__________________________________________________________
Tim Stinchcombe 

Cheltenham, Glos, UK
email: tim102 at tstinchcombe.freeserve.co.uk