Abstract
The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated.
Highlights
Let us start our study by dealing with Problem 1 concerning the conditions under which we can ensure the existence of pseudocoboundary partial Hadamard matrices over a given Latin rectangle L ∈ Rr,n with ns( L) > 0
D L− (S, 1)∆D L− (S, 2) = {2, 4}, which is formed by two elements, as is required by Proposition 7 establishes a lower bound of the cardinality of H( L) for any r × n Latin rectangle L with r > 1 over which a pseudococyclic partial Hadamard matrix associated with the trivial cocycle exists
We have made use of the cocyclic framework over Latin rectangles previously introduced by the authors in [18]
Summary
It is so thatthe existence of coboundaries over non-associative loops has already been proved [17] In this regard, we remind the reader that a cocycle φ over a quasigroup ( Q, ·) is called a coboundary if there exists a map ∂ : Q → {−1, 1} such that φ(i, j) = ∂(i )∂( j)∂(ij), for all i, j ∈ Q. Unlike the cocyclic framework over finite groups, every Goethals–Seidel array constitutes a pseudococyclic Hadamard matrix over a Moufang loop [17]. This last assertion corroborates the relevant role that non-associative quasigroups play in the generalization of the cocyclic framework over groups. Under which conditions may we ensure the existence of a partial Hadamard matrix that is a pseudocoboundary over a given Latin rectangle?.
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