Cross-Correlation Distribution of $p$ -ary $m$-Sequence of Period $p^{4k}-1$ and Its Decimated Sequences by $\left({{ p^{2k}+1}\over { 2}}\right)^{2}$

For an odd prime p, n=4k, and d=((p
^2k
+1)/2)
²
, there are (p
^2k
+1)/2 distinct decimated sequences s(dt+l), 0lesl<(p
^2k
+)/2,of a p-ary m-sequence s(t) of period p
ⁿ
-1 because gcd(d,p
ⁿ
-1)=(p
^2k
+1)/2. In this paper, it is shown that the cross-correlation function between s(t) and s(dt+l),0lesl<(p
^2k
+1)/2, takes the values in {-1, -1 -radic(p
ⁿ
),-1+radic(p
ⁿ
),-1+2radic(p
ⁿ
)} and their cross-correlation distribution is also derived.