Under a mild condition, Ryser\u27s Conjecture holds for every ( n:= 4h^2) with h>1 odd and non square-free

We prove, under a mild condition, that there is no circulant Hadamard matrix ( H) with (n >4) rows when (sqrt{n/4}) is not square-free. The proof introduces a new method to attack Ryser\u27s Conjecture, that is a long standing difficult conjecture

Cocyclic Hadamard matrices over Latin rectangles

In the literature, the theory of cocyclic Hadamard matrices has always been developed over finite groups. This paper introduces the natural generalization of this theory to be developed over Latin rectangles. In this regard, once we introduce the concept of binary cocycle over a given Latin rectangle, we expose examples of Hadamard matrices that are not cocyclic over finite groups but they are over Latin rectangles. Since it is also shown that not every Hadamard matrix is cocyclic over a Latin rectangle, we focus on answering both problems of existence of Hadamard matrices that are cocyclic over a given Latin rectangle and also its reciprocal, that is, the existence of Latin rectangles over which a given Hadamard matrix is cocyclic. We prove in particular that every Latin square over which a Hadamard matrix is cocyclic must be the multiplication table of a loop (not necessarily associative). Besides, we prove the existence of cocyclic Hadamard matrices over non-associative loops of order 2t+3, for all positive integer t > 0.Junta de Andalucía FQM-01