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](http://arxiv.org/abs/1201.4021) and [arxiv:1003.4003, De Launey and Levin](http://arxiv.org/abs/1003.4003), 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. 



Are there infinite constructions for partial circulant hadamard matrices?
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)


