On the Correlation Distribution for a Ternary Niho Decimation

TL;DR

This paper determines the correlation distribution between a ternary m-sequence and its decimation, using weight distribution analysis of a ternary Zetterberg code and solution counting over finite fields, marking a first in non-binary Niho decimation.

Contribution

It provides the first complete correlation distribution for a non-binary Niho decimation sequence, expanding understanding of sequence correlations in finite fields.

Findings

Correlation distribution fully determined for the first time.

Utilizes weight distribution of ternary Zetterberg code.

Counts solutions of equations over finite fields.

Abstract

In this paper, let $n=2m$ and $d=3^{m+1}-2$ with $m\geq2$ and $\gcd(d,3^n-1)=1$. By studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over the finite field $\mathbb{F}_{3^n}$, the correlation distribution between a ternary $m$-sequence of period $3^n-1$ and its $d$-decimation sequence is completely determined. This is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.