Classification of cocyclic Butson Hadamard matrices

Overview

This page contains electronic data associated with the paper Classifying Cocyclic Butson Hadamard matrices by Ronan Egan, Dane Flannery and Padraig Ó Catháin, to which we refer the user for all relevant definitions and for details of the algorithms used to obtain this classification. We provide a complete and irredundant list of representatives of the equivalence classes of cocyclic Butson Hadamard matrices of order n and phase k such that nk < 100. We also provide some MAGMA code for working with such matrices.

Classification

We list only classes of matrices which are non-empty. For non-existence results see the paper. When listing matrices, we give only the exponents of matrix entries relative to some fixed primitive kth root of unity.

• For p < 19, every member of BH(p,p) is equivalent to a Fourier matrix, and hence is cocyclic. For completeness, we include data on these matrices here: BH(3,3), BH(5,5), BH(7,7).

• BH(9,3): up to equivalence there are three cocyclic matrices in this class. One is the tensor product of the matrix of order 3 with itself. The remaining pair are equivalent to circulants, and the members of one class are the Hermitian transposes of the other. For further information click here.

• BH(12,3): up to equivalence there are two classes of cocyclic matrices, the members of one class are the Hermitian transposes of the other. These matrices are equivalent to group-developed matrices. For further information, click here.

• BH(27, 3): there are 16 equivalence classes of cocyclic Butson matrices with these parameters. All are equivalent to group developed matrices. The first and fourth matrices on the list are equivalent to their conjugate transposes. The remainder occur in conjugate-transpose pairs (i, i+1). The matrices have been sorted by the orders of their automorphism groups. For the full listing, click here.

MAGMA files

We include some functions for working with Butson Hadamard and generalised Hadamard matrices in MAGMA. To allow explicit group operations on matrix elements it is convenient to work over the group ring QG. So while we have implemented limited functions for working with Butson Hadamard matrices, automorphism group and equivalence computations take group ring elements as input. Functions for converting one type of matrix to another are given below.

• A MAGMA input file containing all the matrices listed above. These are given as integer matrices, where the entries are the exponent of a primitive root at that co-ordinate. Functions for converting these matrices to Butson or Generalised Hadamard matrices are given below.

• ButsonFunctions.m is a collection of basic functions for working with Butson and generalised Hadamard matrices in MAGMA. In particular, it includes functions for converting generalised Hadamard matrices into Butson matrices, for computing automorphism groups of Butson matrices and for testing equivalence of Butson Hadamard matrices.

• Butson Constructions contains MAGMA implementations of the constructions of Butson and McNulty-Weigert.

Results on McNulty-Weigert matrices

• McNulty and Weigert posed the problem of deciding whether their Butson Hadamard matrix of order 10 over fifth roots of unity is equivalent to the one obtained by Butson's construction for p = 5. The code presented above solves this problem: here is an explicit equivalence.

• The McNulty-Weigert matrix of order 14 is not equivalent to the Butson matrix at this order: the Butson matrix is inequivalent to its conjugate transpose, while the McNulty-Weigert matrix is equivalent to its conjugate transpose. This computation can be carried out explicitly with the code given above.

• From the six MUBS in dimension 5, there are 630 valid choices of matrices for insertion into the McNulty-Weigert array. Of these 390 give a BH(10,10), the remainder give a BH(10, 5). While all BH(10, 5)s are equivalent to the Butson matrix, the BH(10, 10)s appear to be new, and fall into seven equivalence classes. The matrices may be obtained here.

If you find any of the information contained in these pages useful (or if you spot any errors) please let us know. Alternatively, if you would like to cite data or code from this page, please refer to Classifying Cocyclic Butson Hadamard matrices.