On the Correlation Distribution for a Niho Decimation
Yongbo Xia, Nian Li, Xiangyong Zeng, Tor Helleseth · IEEE Transactions on Information Theory · 2017
Let p be a prime, n = 2m and d = 3pm- 2 with m ≥ 2, and gcd(d, pn- 1) = 1. In this paper, the correlation distribution between a p-ary m-sequence of period pn- 1 and its d-decimation sequence is investigated in a unified approach. Some results for the binary case are extended to the general case. It is shown that the problem of determining the correlation distribution for d can be reduced to that of solving two combinatorial problems related to the unit circle of the finite field Fpn. For an arbitrary odd prime p, it seems difficult to solve these two problems. However, for p = 3, by studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over F3n, the two problems are solved, and thus, the corresponding correlation distribution for d is completely determined. It is noteworthy that this is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.