SEQUENCES

The phase modulation and amplitude modulation allow to transmit symbols which are zero or roots of the unity. The autocorrelation function of a sequence $s$ of period $v$ is

$$s\times s(\tau) =\sum_{i=1}^v s_{i+\tau}\,s^\ast_i$$  

$x^\ast$ the conjugate of $x$.

The sequence $s$ is perfect if

$$s\times s(\tau)=\begin{cases}v, &\text{if $\tau=0 \mod v$;}\\0, &\text{else.}\end{cases}$$

[R. Turyn, 1965] For $4<v\leq 12100$, there is no $\pm 1$ perfect sequences.

 
    sequences with small correlations are known : maximal sequences, Legendre sequences...