The existence of multilevel Hadamard matrices (MHMs) of all orders as well as a construction for full-rate circulant MHMs of all orders n 6= 4 is known. We use computer search methods to look for previously unknown full-rate circulant MHMs of orders 5, 6, and 7 and find solutions that potentially do not follow from the known construction. We then give an alternate construction to explain some order six MHMs.
Parker, Keli, "On the Construction of Order Six Multilevel Hadamard Matrices" (2010). Undergraduate Mathematics Day: Proceedings and Other Materials. 26.