On the correlation of subsequences

In 1997 S\'ark\"ozy and Mauduit introduced the well-distribution measure($W$) and the correlation measure of order $\ell$ ($C_{\ell}$) of binarysequences as measures of their pseudorandomness.For a truly random binary sequencethese measures are small ($\ll N^{1/2} (\log N)^c$ for a sequenceof length $N$). Several constructions have been given for which these measuresare small, namely they are $\ll N^{1/2} (\log N)^c$, so the sequence$E_N$ has strong pseudorandom properties. But in certain applications, e.g. incryptography, it is not enough to know that the sequence has strongpseudorandomproperties, it is also important that the subsequences $E_M$ (where $E_M$is of the form$\{e_x,e_{x+1},...,e_{x+M-1}\}$) also have strong pseudorandom propertiesfor values $M$ possibly small in terms of $N$. In this paper I will deal withthis problem incase of values $M \gg N^{1/4+ \varepsilon}$.