70 research outputs found
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
Symmetric Hadamard matrices of orders 268, 412, 436 and 604
We construct many symmetric Hadamard matrices of small order by using the so
called propus construction. The necessary difference families are constructed
by restricting the search to the families which admit a nontrivial multiplier.
Our main result is that we have constructed, for the first time, symmetric
Hadamard matrices of order 268, 412, 436 and 604.Comment: 12 page
On Cocyclic Hadamard Matrices over Goethals-Seidel Loops
About twenty-five years ago, Horadam and de Launey introduced the cocyclic development
of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily
derived. Much more recently, it has been proved that this notion may naturally be extended to
define cocyclic Hadamard matrices developed over a loop. This paper delves into this last topic by
introducing the concepts of coboundary, pseudocoboundary and pseudococycle over a quasigroup,
and also the notion of the pseudococyclic Hadamard matrix. Furthermore, Goethals-Seidel loops
are introduced as a family of Moufang loops so that every Hadamard matrix of Goethals-Seidel type
(which is known not to be cocyclically developed over any group) is actually pseudococyclically
developed over them. Finally, we also prove that, no matter if they are pseudococyclic matrices,
the usual cocyclic Hadamard test is unexpectedly applicable.Junta de AndalucĂa FQM-01
Recent advances in the construction of Hadamard matrices
In the past few years exciting new discoveries have been made in constructing Hadamard matrices. These discoveries have been centred in two ideas:
(i) the construction of Baumert-Hall arrays by utilizing a construction of L. R. Welch, and
(ii) finding suitable matrices to put into these arrays.
We discuss these results, many of which are due to Richard J. Turyn or the author
Goethals--Seidel difference families with symmetric or skew base blocks
We single out a class of difference families which is widely used in some
constructions of Hadamard matrices and which we call Goethals--Seidel (GS)
difference families. They consist of four subsets (base blocks) of a finite
abelian group of order , which can be used to construct Hadamard matrices
via the well-known Goethals--Seidel array. We consider the special class of
these families in cyclic groups, where each base block is either symmetric or
skew. We omit the well-known case where all four blocks are symmetric. By
extending previous computations by several authors, we complete the
classification of GS-difference families of this type for odd . In
particular, we have constructed the first examples of so called good matrices,
G-matrices and best matrices of order 43, and good matrices and G-matrices of
order 45. We also point out some errors in one of the cited references.Comment: 19 pages, including the appendix with a long list of difference
families (about 8 pages
Implementing Hadamard Matrices in SageMath
Hadamard matrices are square matrices with mutually orthogonal
rows. The Hadamard conjecture states that Hadamard matrices of order exist
whenever is , , or a multiple of . However, no construction is
known that works for all values of , and for some orders no Hadamard matrix
has yet been found. Given the many practical applications of these matrices, it
would be useful to have a way to easily check if a construction for a Hadamard
matrix of order exists, and in case to create it. This project aimed to
address this, by implementing constructions of Hadamard and skew Hadamard
matrices to cover all known orders less than or equal to in SageMath, an
open-source mathematical software. Furthermore, we implemented some additional
mathematical objects, such as complementary difference sets and T-sequences,
which were not present in SageMath but are needed to construct Hadamard
matrices.
This also allows to verify the correctness of the results given in the
literature; within the range, just one order, , of a skew
Hadamard matrix claimed to have a known construction, required a fix.Comment: pdflatex+biber, 32 page
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