On the Cross-Correlation of a Ternary $m$-sequence of Period $3^{4k}-1$ and Its Decimated Sequence by $\frac{(3^{2k}+1)^{2}}{20}$

TL;DR

This paper analyzes the cross-correlation between a ternary m-sequence and its decimated version, establishing an upper bound on the correlation magnitude for sequences of period 3^{4k}-1, with implications for sequence design.

Contribution

It provides a new upper bound on the cross-correlation magnitude between a ternary m-sequence and its specific decimated sequence, extending understanding of sequence correlation properties.

Findings

Cross-correlation magnitude is bounded by 5√3^{n}+1.

Sequence period is 3^{4k}-1 with decimation factor ((3^{2k}+1)^2)/20.

The bound applies for odd integer k.

Abstract

Let $d=\frac{(3^{2k}+1)^{2}}{20}$, where $k$ is an odd integer. We show that the magnitude of the cross-correlation values of a ternary $m$-sequence $\{s_{t}\}$ of period $3^{4k}-1$ and its decimated sequence $\{s_{dt}\}$ is upper bounded by $5\sqrt{3^{n}}+1$, where $n=4k$.