On Base, Normal and Near-normal Sequences
Xu Wang, Jiayi Zhu
TL;DR
This paper advances the study of base, normal, and near-normal sequences by introducing a new search algorithm for constructing Base Sequences $BS(n+1,n)$ and providing the first explicit constructions for $n=41,42,43$, thereby confirming the Base Sequence Conjecture up to $n\le 43$. It also analyzes the corresponding conjectures for near-normal and normal sequences, yielding the first counterexamples to the Yang conjecture at $n=42$ and $n=44$ and proving nonexistence of normal sequences for all $n$ of the form $8k-2$, as well as nonexistence for $NS(41)$–$NS(45)$, with $NS(47)$ remaining as the next unknown. The work refines the landscape of Hadamard-construction-related sequences, using modular constraints, norm relations, and backtracking with aggressive pruning to map the boundaries of existence for these sequence families. Overall, the results sharpen the known conditions under which Hadamard matrices can be constructed via Turyn-type sequences and identify precise open cases for future exploration.
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 $n=8k-2,k \in Z_+$.