Binary sequences without optimal autocorrelation

Question

A binary sequence of period $n$ is a sequence $\{s(t)\}$ with $t=0,1,\ldots,n-1$ and $s(t)\in \{0,1\}$. For these sequences define its autocorrelation function in the usual way. For instance, if $n$ is congruent with $0$ modulo $4$, we say that the sequence has optimal autocorrelation if the maximum absolute value of its non-trivial autocorrelations is equal to $4$.

Do binary sequences without optimal autocorrelation have any application or practical importance?

From @kodlu: Inserted from Comment (Why couldn't you do this yourself?). Please insert any information you have about the full distribution of the correlation

$$\begin{array}{|c|c|}\hline n & \text{autocorrelation}\\ \hline 6 & 8 \\ 7 & 16 \\ 8 & 28 \\ 9 & 40\\ \hline \end{array}$$

Answer 1

Yes the sequence has applications in communication theory if all the non-trivial autocorrelation of the sequence are of small value. These sequences are used as preamble for detecting start of a physical layer packet. Packet is a term used for bunch of bits we want to transmit and receive and decode. Lets say we transmit $s(0),...,s(n-1)$ at the beginning of the packet. At receiver we want to correlate the sequence $s(0),...,s(n-1)$ with received sequence and if correlation is maximum we want to declare packet has been received. If the sequence has no optimal "non-trivial" autocorrelation then at the receiver correlation has only one peak so easy to detect and if non-trivial autocorrelation are small, then we can even avoid false packet detection by putting a threshold on all non-trivial autocorrelation to be small.

So if you have such a sequence, you can use it as preamble sequence at the beginning of packet and this sequence can be used to detect the start of the packet. Note that you need to repeat the sequence at the beginning of the packet to make it periodic and then transmit.

Good Luck !

Answer 2

Edit:

Those peak magnitude values are not that great, so any application will depend on the distribution of the correlation. Can you supply more information about the full distribution as well? And do it in the question itself?

The reason is, if those bad values are spread out, i.e., if $|C_s(\omega)|$ is actually small for $\omega$ near zero, say $\omega \in (-T,T) \subseteq \{0\},$ then you have what's called a low correlation zone which means in communication applications such as synchronous CDMA the sequences may have some use. It will also depend on what this small value is, since sequences with autocorrelation zero or at most 1 in the low correlation zone are common.

This question can be answered in two ways. If the alphabet is $\{0,1\}$ and the autocorrelation is defined by $$ \theta(t)=\sum_{k=0}^{N-1} x[k]x[k+t] $$ where the addition in the index is mod $N$ (the period) then this is quite a different situation compared to $$ \theta(t)=\sum_{k=0}^{N-1} (-1)^{x[k]+x[k+t]}. $$ The second is applicable to wi-fi and cellular sequences, preambles, etc. since the actual sequence is transmitted by BPSK. See for example https://en.wikipedia.org/wiki/Maximum_length_sequence.

The first is typically used for optical transmission, also called OOK (on off keying).

You also need to motivate your question further. Are you asking if non-optimal sequences are of practical interest? Theoretical interest? Please put your table into the question and clarify your definition of correlation instead of the vague, "the usual way".