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throughout the question I will talk about sequences in bipolar alphabet, meaning, over {1,-1}.

I have read in the book Shift Register Sequences by Solomon W. Golomb that sequences with the period of $2^n$ can be created by pure cycling register of order n ($PCR_n $). Moreover, it gives some properties about the autocorrelation (that I also read in some papers): $$C(\tau)= \ \begin{cases} 2^n & \tau =0 \\ 0 & 0< \tau \le n-1 \\ \ne0 & \tau =n \end{cases}$$

Another statement (and that is where my question) is that $C(n)$ can bound by the number of cycles of the $PCR_n$ (we can denote the number of cycles as $Z(n)$)

Where this come from I don't really understand.

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You have misunderstood Sol Golomb. A pure cycling register (PCR) of length $n$ generates a sequence of period $n$ and no longer period. It is physically impossible. It just cycles the loaded sequence of length $n$. The sequence it generates may have shorter minimal period, a divisor of $n.$

Take $n=3$ and keep shifting left as in a PCR.

$000\rightarrow 000 \rightarrow 000$ actually has minimal period 1. The same for the loading $111.$

$001\rightarrow 010\rightarrow 100\rightarrow 001$ has minimal period 3. Same with the loading $011.$

The PCR of length 3 has decomposed the space $\{0,1\}^3$ into four cycles. This is because the minimal periods add to the size of the state space: $3+3+1+1=2^3$

Now, he proves this is always the case for a PCR of length $n$ with respect to the state space $\{0,1\}^n,$ and since all this has to do with divisibility relates the number of cycles in the PCR decomposition of length $n,$ which he calls $Z(n)$ to the Euler totient $\phi(n):$ $$ Z(n)=\sum_{d|n} \phi(d)2^{n/d}. $$

Now if you take a maximal length sequence of minimal period $2^n-1$ in the $\pm 1$ formulation and insert a +1, next to the unique run of +1 of length $n-1$ (remember +1 corresponds to 0 mod 2) this PCR decomposition applies. So your statement about $C(\tau)$ is I believe his Theorem 4, p.124, in the reprinted edition of the Aegean Park original.

The autocorrelations of the augmented sequence of period $2^n$ now have an extra term in the sum, and the peak $C(0)=2^n,$ is clear.

Since the way to augment this period is to add a degree $n$ term into the feedback function, and that term is active only once in the period, the autocorrelations at nonzero shifts where $\tau \neq 0\pmod n$ are now 0 instead of -1 for the $2^n-1$ period sequence due to this extra term.

What about when $\tau=n$? Well this won't be zero and getting a hold on its exact value is difficult for a general sequence of period $2^n$, the sequence of theorems 5 through 10 are used here to obtain it for the case that we have an augmented maximal length sequence.

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  • $\begingroup$ I think I got what you are saying, but the question is whether there is a connection between $C(n)$ to $Z(n)$? $\endgroup$ – Mr.O Jun 5 '20 at 11:01
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    $\begingroup$ $Z(n)$ gets large pretty fast with $n$ while $C(n)$ is typically small as Golomb says. I don't know of any other results in this direction, I'll have a look and report if I find anything. $\endgroup$ – kodlu Jun 5 '20 at 11:16
  • $\begingroup$ Maybe it is connected to the fact that DB(n,2) (de bruijn sequence of length $2^n$ over binary {1,0} or {1,-1}) is partitioned into cycles exctly by $PCR_n$? $\endgroup$ – Mr.O Jun 7 '20 at 10:08

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