Why are there no $16$ by $32$ Hadamard circulant matrices? Two rows of a matrix are orthogonal if their inner product equals zero. Call a matrix with all rows pairwise orthogonal an orthogonal matrix. A circulant matrix is one where each row vector is rotated one element to the right relative to the preceding row vector.  We will only consider matrices whose entries are either $-1$ or $1$. 
For number of columns $n= 4,8,12,16,20,24,28, 36$ there exist $n/2$ by $n$ orthogonal circulant matrices.

Why are there no circulant matrices with $16$ rows and $32$ columns which are orthogonal?

Or to phrase it differently, is it possible to prove they don't exist without enumerating them all?
Example 6 by 12 matrix
\begin{pmatrix}
  -1 &\phantom{-}1 &\phantom{-}1 & -1 & -1 &\phantom{-}1 & -1 &\phantom{-}1 & -1 & -1 & -1 & -1\\
  -1 & -1 &\phantom{-}1 &\phantom{-}1 & -1 & -1 &\phantom{-}1 & -1 &\phantom{-}1 & -1 & -1 & -1\\
  -1 & -1 & -1 &\phantom{-}1 &\phantom{-}1 & -1 & -1 &\phantom{-}1 & -1 &\phantom{-}1 & -1 & -1\\
  -1 & -1 & -1 & -1 &\phantom{-}1 &\phantom{-}1 & -1 & -1 &\phantom{-}1 & -1 &\phantom{-}1 & -1\\
  -1 & -1 & -1 & -1 & -1 &\phantom{-}1 &\phantom{-}1 & -1 & -1 &\phantom{-}1 & -1 &\phantom{-}1\\
  \phantom{-}1 & -1 & -1 & -1 & -1 & -1 &\phantom{-}1 &\phantom{-}1 & -1 & -1 &\phantom{-}1 & -1\\\end{pmatrix}
 A: These matrices are known as circulant partial Hadamard matrices and a good reference for these, along with recent results, is $\textit{Circulant partial Hadamard matrices}$ by Craigen, Faucher, Low, and Wares, Lin. Alg. Appl. 439. 
Denote by $r\mbox{-}H(k\times n)$ a $k\times n$ circulant Hadamard matrix in which a row (and hence all) has sum $r$. The authors compile a table of the maximum values of $k$ for $n\le 64$ and all values of $r$. You can see that the $16\times 32$ matrix doesn't exist along with the $22\times 44$ matrix. 
One of the first results in the paper is that if $r\mbox{-}H(k\times n)$ exists then $n$ is divisible by 4. This is why your column numbers are all multiples of 4. Another result is that if Ryser's conjecture is true then $k\le \frac{n}{2}$. The authors show also that there is empirical evidence that the maximum value of $k=\frac{n}{2}$ is attained almost always for $r=2$. A conjecture of Delsarte, Goethals, and Seidel is that a $2\mbox{-}H(k\times 2k)$ exists if and only if $k-1$ is an odd prime power. These two results combined would explain why the $16\times 32$ and $22\times 44$ cases don't exist. It also indicates that the next non-existent case could be $34\times 68$. 
