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.

References

To have a successful construction one must verify that the vector b is indeed the autocorrelation of a binary vector of length 167 and weight 80. Recognizing whether this is the case or not is the central motivational issue for this paper. Currently we simply do not know whether this is true or not, but this article does not center on this issue.

— Convolution numbers: the cyclic case  (2501.18066 - Constantine et al., 30 Jan 2025) in Section 6, final discussion of the Hadamard matrix construction of dimension 668

Resolving this for general s -- or proving that no s achieves mod 64 in this family -- remains an open question.

— An explicit half-flip family of 32-modular Hadamard matrices at L = 3 mod 8: structural placement and mod-tower analysis  (2609.25543 - Kulhandjian, 22 Sep 2026) in Remark following Theorem R.2-tight; Section 6, Open questions