There is Ryser's conjecture (see , page 134): is the inexistence of matrices of order n > 4 that are circulant and Hadamard matrices simultaneously. Now we are interesting in symmetric four- or fifth-circulant Hadamard matrices of rich structure (circulant and back-circulant blocks).
Two four circulant matrices H8
Two four-circulant matrices H12
Two four-circulant matrices H16
Four-circulant matrix H20 and H24
Four-circulant matrices H28
Four-circulant matrices H32 and H36
Fifth-circulant matrix H40 and H48
Four-circulant matrix H52
Four-circulant matrices H60 and H64
Note: matrices H24, H40 have 5 different cells (!)
1. H. J. Ryser, Combinatorial mathematics. The Carus Mathematical Monographs, No. 14, Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York 1963.