Generalized Kasami Sequences: The Large Set

TL;DR

This paper introduces new binary sequence families with large sets and specific correlation properties, generalizing Kasami sequences for even n, with potential applications in communications and cryptography.

Contribution

It constructs new sequence families with optimized correlation properties, extending the Kasami sequence set for even n and various gcd conditions.

Findings

Sequence families have maximum correlation $2^{n/2+1}+1$.

Family size is $2^{3n/2}+2^{n/2}$ or $2^{3n/2}+2^{n/2}-1$ depending on n/2.

Complete correlation distribution is determined.

Abstract

In this paper new binary sequence families $\mathcal{F}^k$ of period $2^n-1$ are constructed for even $n$ and any $k$ with ${\rm gcd}(k,n)=2$ if $n/2$ is odd or ${\rm gcd}(k,n)=1$ if $n/2$ is even. The distribution of their correlation values is completely determined. These families have maximum correlation $2^{n/2+1}+1$ and family size $2^{3n/2}+2^{n/2}$ for odd $n/2$ or $2^{3n/2}+2^{n/2}-1$ for even $n/2$. The construction of the large set of Kasami sequences which is exactly the $\mathcal{F}^{k}$ with $k=n/2+1$ is generalized.