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 [here][1] and [here][2], which address generic partial Hadamard matrices, as opposed to those that are partial circulant.


  [1]: http://arxiv.org/abs/1201.4021
  [2]: http://arxiv.org/abs/1003.4003