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Padraig Ó Catháin

Publications and source records attributed to Padraig Ó Catháin.

At least 19 recordsLinked to original sources
arXiv2026

Complex generalised weighing matrices in centraliser algebras of monomial representations

An $n \times n$ matrix $W$ with exactly $w$ non-zero entries taken from the set of $k^{\rm th}$ complex roots of unity in each row and column satisfying $WW^{\ast} = wI_n$ is a complex generalised weighing matrix $CGW(n,w;k)$. We study such matrices through the centraliser algebras of monomial representations of finite groups. Using an exhaustive search over the linear characters of Schur covers, we classify, up to monomial equivalence, the complex generalised weighing matrices admitting a primitive group of rank at most five and degree at most $80$ acting by strong automorphisms, for coefficient orders $k \leq 6$, with partial results for larger degrees $100$. The census recovers known infinite families related to projective and affine finite geometries, describes infinite families related to Hamming schemes and settles the existence of some small open cases enumerated in the literature. We construct quantum error-correcting codes from the these matrices and determine their minimum distances exactly in all cases.

math.CO↗
arXiv2026

Symmetries of regular $q$-graphs

Given a finite vector space $V=\mathbb{F}_q^n$, the $q$-analogue of a graph, called a $q$-graph, is a pair $Γ=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of $1$-dimensional subspaces of $V$ and $\mathcal{E}$ is a subset of the $2$-dimensional subspaces of $V$. Elements of $\mathcal{V}$ and $\mathcal{E}$ are called vertices and edges, respectively. If the edges through a vertex $X$ consist of all $2$-spaces of a $(k+1)$-dimensional space which contain $X$, regardless of the choice of vertex, then $Γ$ is $k$-regular. Moreover, $Γ$ is flag-transitive if there is a subgroup of $Γ{\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all incident vertex-edge pairs; and symmetric if there is a subgroup of $Γ{\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all ordered pairs of adjacent vertices. This paper classifies all $k$-regular $q$-graphs that are either flag-transitive or symmetric. The $q$-graphs in the classification are constructed from familiar objects in finite geometry, including spreads, symplectic polar spaces, and generalised hexagons. The classification depends essentially on the classification of transitive linear groups, and thus ultimately on the classification of finite simple groups.

math.CO↗
arXiv2025

Centraliser algebras of monomial representations and applications in combinatorics

Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. G. Higman and others, we give an explicit construction for a basis of the centraliser algebra of a monomial representation. The character table of this algebra is then constructed via character sums over double cosets. We locate the theory of group-developed and cocyclic-developed Hadamard matrices within this framework. We apply Gröbner bases to produce a new classification of highly symmetric complex Hadamard matrices.

math.CO↗
arXiv2025

A generalisation of bent vectors for Butson Hadamard matrices

An $n\times n$ complex matrix $M$ with entries in the $k^{\textrm{th}}$ roots of unity which satisfies $MM^{\ast} = nI_{n}$ is called a Butson Hadamard matrix. While a matrix with entries in the $k^{\textrm{th}}$ roots typically does not have an eigenvector with entries in the same set, such vectors and their generalisations turn out to have multiple applications. A bent vector for $M$ satisfies $M{\bf x} = λ{\bf y}$ where ${\bf x}$ has entries in the $k^{\textrm{th}}$ roots of unity and all entries of $\textbf{y}$ are complex numbers of norm $1$. Such a bent vector ${\bf x}$ is self-dual if ${\bf y} = μ{\bf x}$ and conjugate self-dual if ${\bf y} = μ\overline{\bf x}$ for some $μ$ of norm $1$. Using techniques from algebraic number theory, we prove some order conditions and non-existence results for self-dual and conjugate self-dual bent vectors; using tensor constructions and Bush-type matrices we give explicit examples. We conclude with an application to the covering radius of certain non-linear codes generalising the Reed Muller codes.

math.CO↗
arXiv2025

Extendibility of Latin Hypercuboids

A Latin hypercuboid of order $n$ is a $d$-dimensional matrix of dimensions $n\times n\times\cdots\times n\times k$, with symbols from a set of cardinality $n$ such that each symbol occurs at most once in each axis-parallel line. If $k=n$ the hypercuboid is a Latin hypercube. The Latin hypercuboid is \emph{completable} if it is contained in a Latin hypercube of the same order and dimension. It is \emph{extendible} if it can have one extra layer added. In this note we consider which Latin hypercuboids are completable/extendible. We also consider a generalisation that involves multidimensional arrays of sets that satisfy certain balance properties. The extendibility problem corresponds to choosing representatives from the sets in a way that is analogous to a choice of a Hall system of distinct representatives, but in higher dimensions. The completability problem corresponds to partitioning the sets into such SDRs. We provide a construction for such an array of sets that does not have the property analogous to completability. A related concept was introduced by Häggkvist under the name $(m,m,m)$-array. We generalise a construction of $(m,m,m)$-arrays credited to Pebody, but show that it cannot be used to build the arrays that we need.

math.CO↗
arXiv2024

Triangle-free graphs with diameter 2

There are finitely many graphs with diameter $2$ and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter $2$ and no $K_{2,3}$ subgraph? This question is related to the existence of triangle-free strongly regular graphs, but allowing for a range of co-degrees gives the question a more extremal flavour. More generally, for fixed $s$ and $t$, are there infinitely many twin-free triangle-free $K_{s,t}$-free graphs with diameter 2? This paper presents partial results regarding these questions, including computational results, potential Cayley-graph and probabilistic constructions.

math.CO↗
arXiv2024

A Bollobás-type problem: from root systems to Erdős-Ko-Rado

Motivated by an Erdős--Ko--Rado type problem on sets of strongly orthogonal roots in the $A_{\ell}$ root system, we estimate bounds for the size of a family of pairs $(A_{i}, B_{i})$ of $k$-subsets in $\{ 1, 2, \ldots, n\}$ such that $A_{i} \cap B_{j}= \emptyset$ and $|A_{i} \cap A_{j}| + |B_{i} \cap B_{j}| = k$ for all $i \neq j$. This is reminiscent of a classic problem of Bollobás. We provide upper and lower bounds for this problem, relying on classical results of extremal combinatorics and an explicit construction using the incidence matrix of a finite projective plane.

math.CO↗
arXiv2023

Segre's theorem on ovals in Desarguesian projective planes

Segre's theorem on ovals in projective spaces is an ingenious result from the mid-twentieth century which requires surprisingly little background to prove. This note, suitable for undergraduates with experience of linear and abstract algebra, provides a complete and self-contained proof. All necessary pre-requisites, principally evaluation of homogeneous polynomials at projective points and Desargues' theorem are presented in full. While following the broad outline of Segre's proof, careful parameterisation of certain tangent lines results in shorter and simpler computations than the original.

math.CO↗
arXiv2022

Block avoiding point sequencings of partial Steiner systems

A partial $(n,k,t)_λ$-system is a pair $(X,\mathcal{B})$ where $X$ is an $n$-set of vertices and $\mathcal{B}$ is a collection of $k$-subsets of $X$ called blocks such that each $t$-set of vertices is a subset of at most $λ$ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of $\{0,\ldots,n-1\}$. A sequencing is $\ell$-block avoiding or, more briefly, $\ell$-good if no block is contained in a set of $\ell$ vertices with consecutive labels. Here we give a short proof that, for fixed $k$, $t$ and $λ$, any partial $(n,k,t)_λ$-system has an $\ell$-good sequencing for some $\ell=Θ(n^{1/t})$ as $n$ becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case $k=t+1$ where results of Kostochka, Mubayi and Verstraëte show that the value of $\ell$ cannot be increased beyond $Θ((n \log n)^{1/t})$. A special case of our result shows that every partial Steiner triple system (partial $(n,3,2)_1$-system) has an $\ell$-good sequencing for each positive integer $\ell \leq 0.0908\,n^{1/2}$.

math.CO↗
arXiv2020

Morphisms of Skew Hadamard Matrices

Quaternary unit Hadamard (QUH) matrices were introduced by Fender, Kharagani and Suda along with a method to construct them at prime power orders. We present a novel construction of real Hadamard matrices from QUH matrices. Our construction recovers the result by Mukhopadhyay on the existence of real Hadamard matrices of order $q^n+q^{n-1}$ for each prime power $q\equiv 3\mod 4$. Furthermore we provide nonexistence conditions for QUH matrices.

math.CO↗
arXiv2019

Dimensions of semi-simple matrix algebras

For $n \geq 225$ we show that every integer of the form $n + 2m$ such that $0 \leq 2m \leq n^{2} - \frac{9}{2} n \sqrt{n}$ is the dimension of a connected semi-simple subalgebra of $\mathrm{M}_{n}(k)$, that is, a subalgebra isomorphic to a direct sum of $t$ disjoint subalgebras $\mathrm{M}_{n_{i}}(k)$, where $\sum_{i=1}^{t} n_{i} = n$. From this, we conclude that the density of integers in $[0,\ldots, n^{2}]$ which are the dimension of a semi-simple subalgebra of $\mathrm{M}_{n}(k)$ tends to $1$ as $n \rightarrow \infty$.

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arXiv2018

Sparsification of Matrices and Compressed Sensing

Compressed sensing is a signal processing technique whereby the limits imposed by the Shannon--Nyquist theorem can be exceeded provided certain conditions are imposed on the signal. Such conditions occur in many real-world scenarios, and compressed sensing has emerging applications in medical imaging, big data, and statistics. Finding practical matrix constructions and computationally efficient recovery algorithms for compressed sensing is an area of intense research interest. Many probabilistic matrix constructions have been proposed, and it is now well known that matrices with entries drawn from a suitable probability distribution are essentially optimal for compressed sensing. Potential applications have motivated the search for constructions of sparse compressed sensing matrices (i.e., matrices containing few non-zero entries). Various constructions have been proposed, and simulations suggest that their performance is comparable to that of dense matrices. In this paper, extensive simulations are presented which suggest that sparsification leads to a marked improvement in compressed sensing performance for a large class of matrix constructions and for many different recovery algorithms.

cs.IT↗
arXiv2017

Morphisms of Butson classes

We introduce the concept of a morphism from the set of Butson Hadamard matrices over kth roots of unity to the set of Butson matrices over $\ell$th roots of unity. As concrete examples of such morphisms, we describe tensor-product-like maps which reduce the order of the roots of unity appearing in a Butson matrix at the cost of increasing the dimension. Such maps can be constructed from Butson matrices with eigenvalues satisfying certain natural conditions. Our work unifies and generalises Turyn's construction of real Hadamard matrices from Butson matrices over the 4th roots and the work of Compton, Craigen and de Launey on `unreal' Butson matrices over the 6th roots. As a case study, we classify all morphisms from the set of $n \times n$ Butson matrices over kth roots of unity to the set of $2n\times 2n$ Butson matrices over $\ell$th roots of unity where $\ell < k$.

math.CO↗
arXiv2017

Improved User-Private Information Retrieval via Finite Geometry

In a User-Private Information Retrieval (UPIR) scheme, a set of users collaborate to retrieve files from a database without revealing to observers which participant in the scheme requested the file. Protocols have been proposed based on pairwise balanced designs and symmetric designs. Wepropose a new class of UPIR schemes based on generalised quadrangles (GQ). We prove that while the privacy of users in the previously proposed schemes could be compromised by a single user, the new GQ-UPIR schemes proposed in this paper maintain privacy with high probability even when up to $O(n^{1/4 - ε})$ users collude, where $n$ is the total number of users in the scheme.

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arXiv2016

Explicit correlation amplifiers for finding outlier correlations in deterministic subquadratic time

We derandomize G. Valiant's [J. ACM 62 (2015) Art. 13] subquadratic-time algorithm for finding outlier correlations in binary data. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant's randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders in Reingold, Vadhan, and Wigderson [Ann. of Math. 155 (2002) 157--187]. We say that a function $f:\{-1,1\}^d\rightarrow\{-1,1\}^D$ is a correlation amplifier with threshold $0\leqτ\leq 1$, error $γ\geq 1$, and strength $p$ an even positive integer if for all pairs of vectors $x,y\in\{-1,1\}^d$ it holds that (i) $|\langle x,y\rangle|<τd$ implies $|\langle f(x),f(y)\rangle|\leq(τγ)^pD$; and (ii) $|\langle x,y\rangle|\geqτd$ implies $\bigl(\frac{\langle x,y\rangle}{γd}\bigr)^pD \leq\langle f(x),f(y)\rangle\leq \bigl(\frac{γ\langle x,y\rangle}{d}\bigr)^pD$.

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arXiv2016

Remark on a result of Constantine

In this short note we construct codes of length $4n$ with $8n+8$ codewords and minimum distance $2n-2$ whenever $4n+4$ is the order of a Hadamard matrix. This generalises work of Constantine who obtained a similar result in the special case that $n$ is a prime power.

math.CO↗
arXiv2015

Compressed sensing with combinatorial designs: theory and simulations

In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the performance of matrices obtained from this construction. We provide new theoretical results and detailed simulations. These simulations 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.

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