5 research outputs found
Invariants of Quadratic Forms and applications in Design Theory
The study of regular incidence structures such as projective planes and
symmetric block designs is a well established topic in discrete mathematics.
Work of Bruck, Ryser and Chowla in the mid-twentieth century applied the
Hasse-Minkowski local-global theory for quadratic forms to derive non-existence
results for certain design parameters. Several combinatorialists have provided
alternative proofs of this result, replacing conceptual arguments with
algorithmic ones. In this paper, we show that the methods required are purely
linear-algebraic in nature and are no more difficult conceptually than the
theory of the Jordan Canonical Form. Computationally, they are rather easier.
We conclude with some classical and recent applications to design theory,
including a novel application to the decomposition of incidence matrices of
symmetric designs.Comment: 23 page
A Survey of the Hadamard Maximal Determinant Problem
In a celebrated paper of 1893, Hadamard established the maximal determinant
theorem, which establishes an upper bound on the determinant of a matrix with
complex entries of norm at most . His paper concludes with the suggestion
that mathematicians study the maximum value of the determinant of an matrix with entries in . This is the Hadamard maximal
determinant problem.
This survey provides complete proofs of the major results obtained thus far.
We focus equally on upper bounds for the determinant (achieved largely via the
study of the Gram matrices), and constructive lower bounds (achieved largely
via quadratic residues in finite fields and concepts from design theory). To
provide an impression of the historical development of the subject, we have
attempted to modernise many of the original proofs, while maintaining the
underlying ideas. Thus some of the proofs have the flavour of determinant
theory, and some appear in print in English for the first time.
We survey constructions of matrices in order , giving
asymptotic analysis which has not previously appeared in the literature. We
prove that there exists an infinite family of matrices achieving at least
of the maximal determinant bound. Previously the best known constant for
a result of this type was .Comment: 28 pages, minor modification
Compressed sensing with combinatorial designs: theory and simulations
We use deterministic and probabilistic methods to analyze the performance of compressed sensing matrices constructed from Hadamard matrices and pairwise balanced designs, previously introduced by a subset of the authors. In this paper, we obtain upper and lower bounds on the sparsity of signals for which our matrices guarantee recovery. These bounds are tight to within a multiplicative factor of at most. We provide new theoretical results and detailed simulations, which indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given