Cocyclic Hadamard matrices over Latin rectangles

Abstract 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 2 t + 3 , for all positive integer t > 0 .

[1]  M. Hall.  An existence theorem for latin squares , 1945 .

[2]  D. Flannery,et al.  Cocyclic Hadamard Matrices and Hadamard Groups Are Equivalent , 1997 .

[3]  V. Álvarez,et al.  On D4t ‐Cocyclic Hadamard Matrices , 2016 .

[4]  K. Horadam.  Hadamard Matrices and Their Applications , 2006 .

[5]  K. J. Horadam,et al.  Generation of Cocyclic Hadamard Matrices , 1995 .

[6]  K. J. Horadam,et al.  Cocyclic Development of Designs , 1993 .

[7]  B. McKay,et al.  Small latin squares, quasigroups, and loops , 2007 .

[8]  J. J. Seidel,et al.  SYMMETRIC HADAMARD MATRICES OF ORDER 36 , 1970 .

[9]  A. Krapez,et al.  Belousov equations on quasigroups , 1987 .

[10]  Bernhard Schmidt,et al.  The anti-field-descent method , 2016, J. Comb. Theory, Ser. A.

[11]  Padraig Ó Catháin,et al.  The cocyclic Hadamard matrices of order less than 40 , 2011, Des. Codes Cryptogr..

[12]  J. Sylvester.  LX. Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers , 1867 .

[13]  R. Turyn.  Character sums and difference sets. , 1965 .

[14]  Patric R. J. Östergård,et al.  The number of Latin squares of order 11 , 2009, Math. Comput..

[15]  Brendan D. McKay,et al.  Isomorph-Free Exhaustive Generation , 1998, J. Algorithms.

[16]  A. Krapez,et al.  Quasigroups satisfying balanced but not Belousov equations are group isotopes , 1991 .

[17]  Clement W. H. Lam,et al.  On the number of 8×8 latin squares , 1990, J. Comb. Theory, Ser. A.

[18]  J. J. Seidel,et al.  Strongly Regular Graphs Derived from Combinatorial Designs , 1970, Canadian Journal of Mathematics.

[19]  R. Moufang,et al.  Zur Struktur von Alternativkörpern , 1935 .

[20]  J. Dénes,et al.  Latin squares and their applications , 1974 .

[21]  Kathy J. Horadam,et al.  A weak difference set construction for higher dimensional designs , 1993, Des. Codes Cryptogr..

[22]  Michael J. Mossinghoff,et al.  Wieferich pairs and Barker sequences, II , 2014 .