A new Yang number and consequences

Abstract

Base sequences BS(m, n) are quadruples (A; B; C; D) of {±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 (Ðoković, Classification of near-normal sequences, arXiv:0903.4390v1 [math.CO] 25 Mar (2009); Ðoković, Some new near-normal sequences, arXiv:0907.3129v1 [math.CO] 17 Jul (2009)) 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.