On higher dimensional cocyclic Hadamard matrices

Abstract

Provided that a cohomological model for G is known, we describe a method for constructing a basis for n-cocycles over G, from which the whole set of n-dimensional n-cocyclic matrices over G may be straightforwardly calculated. Focusing in the case n = 2 (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative 2-cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When n > 2, this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for n = 2, 3. In particular, we give some examples of improper 3-dimensional 3-cocyclic Hadamard matrices. Keywords: (co)homological model cocyclic matrix proper/improper higher dimensional Hadamard matrix.

Cubes of symmetric designs

“…The design D 3 is obtained by applying another switch to D 2 . Take S 2 = {g 6 , g 7 , g 14 , g 15 } and replace the blocks The smallest parameters of symmetric designs are (7, 3, 1), (11,5,2), and (13, 4, 1). There is a single design up to isomorphism and duality for each of these parameters, coming from a difference set in the cyclic group Z v .…”

“…Among others, it encompasses symmetric block designs, Hadamard matrices, and generalisations such as orthogonal designs and weighing matrices. While n-dimensional Hadamard matrices have been studied before [3,4,15,16,31,32,33,35,36,37] and after [5,10,12,13,26] de Launey's paper, and are even featured in books [1, §6], [2, Chapter 10], [11,Chapter 11], [18,Chapter 5], [38], there seem to be no works dedicated to n-dimensional symmetric designs.…”

“…Finally, Section 5 is devoted to computational results about cubes with small parameters. A complete classification of group cubes is performed for (16,6,2) and (21,5,1). Examples of non-difference group cubes are found for (27,13,6), (36,15,6), (45, 12, 3), (63,31,15), (64,28,12), and (96, 20,4).…”

Quantum LDPC Codes Based on Cocyclic Block Matrices

“…here, G x and G z are generator matrices of the encoding bit and phase, respectively. Since a cocyclic approach was introduced in [24], some interesting codes come from useful cocyclic matrices, such as the Hadamard matrix that is the matrix representation of a cocycle [25].…”

“…Based on the current research results in this field, we will further study the encoding methods of quantum long-codes stabilizers. In this paper, a class of quantum stabilizer codes spirited with a family of cocyclic matrices is proposed [ 24 , 25 ]. The involved cocyclic block matrices are based on the Jacket matrix, of which the main property is that its inverse matrix can be gained by block-wise inverse [ 26 , 27 ].…”