On Correlation Distribution of Niho-Type Decimation d = 3(pm - 1) + 1

Abstract

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.