On Correlation Distribution of Niho-Type Decimation <i>d</i> = 3(<i>p^m </i> - 1) + 1

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d=3(p^{m}-1)+1$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(p^{2m})$ </tex-math></inline-formula> for any prime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p \ge 5$ </tex-math></inline-formula>. Previously this problem was solved by Xia et al. only for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p=2$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p=3$ </tex-math></inline-formula> in a series of papers. The main difficulty of this problem for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p \ge 5$ </tex-math></inline-formula>, 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p \ge 7$ </tex-math></inline-formula>, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem.