An Open Problem on the Distribution of a Niho-Type Cross-Correlation Function

Abstract

In this paper, let $n=2k$ and $d=3\cdot 2^{k}-2$ with $k\geq 3$ and $\gcd (d,2^{n}-1)=1$ . Based on some analysis of certain equations over finite fields and the number of codewords with Hamming weight five in Zetterberg code, the correlation distribution between a binary $m$ -sequence of period $2^{n}-1$ and its $d$ -decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al.