Cross-Correlation Distribution between a p-Ary m-Sequence and Its Decimated Sequence with Decimation Factor $d=\frac{(p^{m}+1)^2}{2(p^e+1)}$

Abstract

Let p be an odd prime and m be any positive integer. Assume that n=2m and e is a positive divisor of m with m/e being odd. For the decimation factor $d=\frac{(p^{m}+1)^2}{2(p^e+1)}$, the cross-correlation between the p-ary m-sequence $\{tr_1^n(\alpha^i)\}$ and its decimated sequence $\{tr_1^n(\alpha^{di})\}$ is investigated. The value distribution of the correlation function is completely determined. The results in this paper generalize the previous results given by Choi, Luo and Sun et al., where they considered some special cases of the decimation factor d with a restriction that m is odd. Note that the integer m in this paper can be even or odd. Thus, the decimation factor d here is more flexible than the previous ones. Moreover, our method for determining the value distribution of the correlation function is different from those adopted by Luo and Sun et al. in that we do not need to calculate the third power sum of the correlation function, which is usually difficult to handle.