A new Yang number and consequences

TL;DR

This paper extends the classification of near-normal sequences to s=34, constructs NN(36), and introduces T-sequences of length 73, establishing 73 as a Yang number with significant implications.

Contribution

It provides the first classification of NN(34), constructs NN(36), and produces T-sequences of length 73, revealing new Yang numbers and their consequences.

Findings

Classified NN(s) for s=34

Constructed NN(36)

Produced T-sequences of length 73

Abstract

Base sequences BS(m,n) are quadruples (A;B;C;D) of {+1,-1}-sequences, A and B of length m and C and D of length n, the sum of whose non-periodic auto-correlation functions is zero. Base sequences and some special subclasses of BS(n+1,n) known as normal and near-normal sequences, NS(n) and NN(n), as well as T-sequences and orthogonal designs play a prominent role in modern constructions of Hadamard matrices. In our previous papers (see the references) we have classified the near-normal sequences NN(s) for all even integers s <= 32 (they do not exist for odd s>1). We now extend the classification to the case s=34. Moreover we construct the first example of near-normal sequences NN(36). Consequently, we construct for the first time T-sequences of length 73. For all smaller lengths, T-sequences were already known. Another consequence is that 73 is a Yang number, and a few important consequences of this fact are given.