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The component-wise or Schur product $C*C'$ of two linear error correcting codes $C$ and $C'$ over certain finite field is the linear code spanned by all component-wise products of a codeword in $C$ with a codeword in $C'$. When $C=C'$, we…
By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length…
Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and…
A class of $(2n)^2\times(2n)^2$ multiparameter braid matrices are presented for all $n$ $(n\geq 1)$. Apart from the spectral parameter $\theta$, they depend on $2n^2$ free parameters $m_{ij}^{(\pm)}$, $i,j=1,...,n$. For real parameters the…
A new construction of complex Hadamard matrices of composite order d=pq, with primes p,q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the…
We provide a geometric characterization of $k$-dimensional $\mathbb{F}_{q^m}$-linear sum-rank metric codes as tuples of $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We then use this characterization to study one-weight codes in the…
The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Pl\"ucker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a…
The generalized Hamming weights (GHWs) of a linear code C extend the concept of minimum distance, which is the minimum cardinality of the support of all one-dimensional subspaces of C, to the minimum cardinality of the support of all…
Given graphs $G$ and $H$, we say that $G$ is $H$-$good$ if the Ramsey number $R(G,H)$ equals the trivial lower bound $(|G| - 1)(\chi(H) - 1) + \sigma(H)$, where $\chi(H)$ denotes the usual chromatic number of $H$, and $\sigma(H)$ denotes…
We introduce a new fundamental algorithm called Matrix-POAFD to solve the matrix least square problem. The method is based on the matching pursuit principle. The method directly extracts, among the given features as column vectors of the…
This preprint is of a chapter to appear in {\it Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and Applied Mathematics}, K.-U. Schmidt and A. Winterhof (eds.).…
We review a method to obtain optimal Poincar\'e-Hardy-type inequalities on the hyperbolic spaces, and discuss briefly generalisations to certain classes of Riemannian manifolds. Afterwards, we recall a corresponding result on homogeneous…
We present a preliminary study of Schur norms $\|M\|_{S}=\max\{ \|M\circ C\|: \|C\|=1\}$, where M is a matrix whose entries are $\pm1$, and $\circ$ denotes the entrywise (i.e., Schur or Hadamard) product of the matrices. We show that, if…
We consider optimization problems involving the multiplication of variable matrices to be selected from a given family, which might be a discrete set, a continuous set or a combination of both. Such nonlinear, and possibly discrete,…
For a matrix $M$ and a positive integer $r$, the rank $r$ rigidity of $M$ is the smallest number of entries of $M$ which one must change to make its rank at most $r$. There are many known applications of rigidity lower bounds to a variety…
Binary duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root bound. In this paper, we present several families of binary duadic codes of length $2^m-1$…
We introduce the n-th Heron variety as the realization space of the (squared) volumes of faces of an n-simplex. Our primary goal is to understand the extent to which Heron's formula, which expresses the area of a triangle as a function of…
In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $\mathcal{H}$ in $K_n$, defined as the minimum number of colours needed to colour the edges of $K_n$ so that every copy of a graph $H\in…
We analyze the problem of folding one polyhedron, viewed as a metric graph of its edges, into the shape of another, similar to 1D origami. We find such foldings between all pairs of Platonic solids and prove corresponding lower bounds,…
We consider the problem of deriving upper bounds on the parameters of sum-rank-metric codes, with focus on their dimension and block length. The sum-rank metric is a combination of the Hamming and the rank metric, and most of the available…