On Base, Normal and Near-normal Sequences

TL;DR

This paper advances the understanding of base, normal, and near-normal sequences by constructing new cases, providing counterexamples to existing conjectures, and analyzing their existence for various lengths.

Contribution

It presents algorithms for constructing BS(n+1,n) for n=41,42,43, and identifies counterexamples to Yang's conjecture for NNS(n), also proving non-existence of NS(n) for certain lengths.

Findings

Constructed BS(n+1,n) for n=41,42,43.

Discovered no NNS(n) for n=42,44, countering Yang's conjecture.

Proved no NS(n) exists for n=8k-2, k in Z+.

Abstract

The base sequences BS(n+1,n) are four sequences of $\pm1$ and lengths n+1,n+1,n,n with zero auto correlation. The base sequence conjecture states that BS(n+1,n) exists for all positive integers and has been verified for $n\le40$. We present our algorithm and give construction of BS(n+1,n) for $n=41,42,43$.\\ The Normal sequences NS (n) and the Near-normal sequences NNS (n) are subclasses of BS(n+1,n). Yang conjecture asserts that there is a NNS(n) for each even integer n and has been verified for $n\le40$. We found that there is no NNS(n) for n=42 and 44 by exhaustive search, which gives the first counter case of Yang conjecture. We also show that there is no NS(n) for n=41,42,43,44,45 by exhaustive search and proves that no NS(n) exist for $n=8k-2,k \in Z_+$.