The cross-correlation problem is a classic problem in sequence design. In this paper we compute the cross-correlation distribution of the Niho-type decimation $d=3(p^m-1)+1$ over $\mathrm{GF}(p^{2m})$ for any prime $p \ge 5$. Previously this problem was solved by Xia et al. only for $p=2$ and $p=3$ in a series of papers. The main difficulty of this problem for $p \ge 5$, as pointed out by Xia et al., is to count the number of codewords of "pure weight" 5 in $p$-ary Zetterberg codes. It turns out this counting problem can be transformed by the MacWilliams identity into counting codewords of weight at most 5 in $p$-ary Melas codes, the most difficult of which is related to a K3 surface well studied in the literature and can be computed. When $p \ge 7$, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem.

翻译：互相关问题是序列设计中的经典问题。本文针对任意素数$p \ge 5$，计算了$\mathrm{GF}(p^{2m})$上Niho型抽取$d=3(p^m-1)+1$的互相关分布。此前，Xia等人在一系列论文中仅解决了$p=2$和$p=3$的情形。如Xia等人所指出的，对于$p \ge 5$的情形，该问题的主要难点在于计算$p$进制Zetterberg码中"纯重量"为5的码字数目。通过MacWilliams恒等式，该计数问题可转化为计算$p$进制Melas码中重量不超过5的码字数目，而其中最困难的情形与文献中已深入研究的K3曲面相关，可通过计算求解。当$p \ge 7$时，有限域上的椭圆曲线理论也在该问题的解决过程中发挥了重要作用。