Are there infinite constructions for partial circulant Hadamard matrices?

Question

I believe that the circulant Hadamard conjecture (that there are no circulant Hadamard matrices of size greater than $4\times4$) is still open.

I also know that examples of $(n/2) \times n$ matrices which are partial Hadamard circulant have been found experimentally for moderate values of $n.$

To clarify, a partial circulant $m(n)\times n$ Hadamard matrix $A$ has entries from $\{-1,+1\}$ and if its first row is $a,$ then its subsequent rows are $T(a),\ldots,T^{m(n)-1}(a)$ where $T$ denotes, say, the left cyclic shift operator and $T^k$ is $T$ composed with itself $k$ times.

Is there a known infinite construction for partial circulant Hadamard matrices? Equivalently does a sequence of $m_k(n_k)\times n_k$ Hadamard matrices exist, where $n_k\rightarrow \infty$ and so does $m_k(n_k)$. Note that $m_k(n_k)=o(n_k)$ is allowed, under this definition.

I am aware of references arXiv:1201.4021, Armario et al and arxiv:1003.4003, De Launey and Levin, which address generic partial Hadamard matrices, as opposed to those that are partial circulant.

Edit: To make things explicit, the best results I have found without the circulant constraint are the following: For any $\varepsilon>0$ and for $n$ large enough, $t\equiv 0~(mod~4)$ there is a $n\times t$ partial Hadamard matrix if $n\leq \frac{t}{2}-t^{\frac{113}{132}+\varepsilon}.$ Subject to the extended Riemann hypothesis the $\frac{113}{132}$ in the exponent can be replaced by $\frac{7}{12}.$ This matches computational results, see https://codegolf.stackexchange.com/questions/55726/an-optimization-version-of-the-hadamard-problem, that have found $\frac{n}{2}\times n$ partial Hadamard matrices up to $n$ somewhere between 50 and 100.

Answer 1

Let $r\mbox{-}H(k\times n)$ denote a $k\times n$ partial circulant Hadmard matrix in which a row (and hence all) sums to $r$. It's a known result, see Theroem 9 in [1], that $2\mbox{-}H((p+1)\times 2(p+1))$ exists for any prime power $p$. This is because negacyclic $C$-matrices of order $p+1$ exist. In [2], Paley gave a construction of $C$-matrices using the Legendre symbol $\chi$ of the Galois field GF$(p)$. A variation of this construction leads to a negacyclic form for these Paley matrices, see [3] for details.

[1] $\textit{Circulant partial Hadamard matrices}$ by Craigen, Faucher, Low, and Wares, Lin. Alg. Appl. 439 (2013)

[2] $\textit{On orthogonal matrices}$ by Paley, J. Math. Phys. 12 (1933)

[3] $\textit{Orthogonal matrices with zero diagonal II}$ by Delsarte, Goethals, and Seidel, Can. J. Math., Vol. XXIII No. 5 (1971)