Autocorrelation realization of the 167-term vector b

Determine whether the explicitly displayed 83-component vector b in Section 6 is the autocorrelation vector of a binary vector of length 167 and weight 80, thereby determining whether the proposed Goethals–Seidel construction can be completed for a Hadamard matrix of dimension 668.

Background

The paper discusses the Goethals–Seidel construction of Hadamard matrices from four binary vectors of length n whose cyclic autocorrelations sum componentwise to a prescribed constant. For dimension 668, corresponding to n=167, the authors identify binary vectors of weights 76, 76, and 77 whose autocorrelations leave the displayed vector b as the required autocorrelation of a fourth binary vector of weight 80.

Existence of such a fourth binary vector would complete the construction of a Hadamard matrix of dimension 668. The paper explicitly states that it is currently unknown whether b is realizable as the autocorrelation vector of a binary vector with the required length and weight, while noting that this question is not the article’s central focus.