65 research outputs found
On weighing matrices with square weights
We give a new construction for a known family of weighing matrices using the 2-adjugate method of Vartak and Patwardhan. We review the existence of W(n,k²), k = 1,.. ,12, giving new results for k = 8,...12
Exhaustive Search for Small Dimension Recursive MDS Diffusion Layers for Block Ciphers and Hash Functions
This article presents a new algorithm to find MDS matrices that are well
suited for use as a diffusion layer in lightweight block ciphers. Using an
recursive construction, it is possible to obtain matrices with a very compact
description. Classical field multiplications can also be replaced by simple
F2-linear transformations (combinations of XORs and shifts) which are much
lighter. Using this algorithm, it was possible to design a 16x16 matrix on a
5-bit alphabet, yielding an efficient 80-bit diffusion layer with maximal
branch number.Comment: Published at ISIT 201
Parity of transversals of Latin squares
We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order , the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving , where is the number of diagonals of a given
Latin square that contain exactly different symbols.
Let denote the matrix obtained by deleting row and column
from a parent matrix . Define to be the number of transversals
in , for some fixed Latin square . We show that for all and . Also, if has odd order then the
number of transversals of equals mod 2. We conjecture that for all .
In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a -regular bipartite graph on vertices is divisible by
when is odd and . We also show that for all , when is an integer matrix of odd
order with all row and columns sums equal to
On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix
For a positive integer, we find restrictions modulo on the
coefficients of the characteristic polynomial of a Seidel matrix
. We show that, for a Seidel matrix of order even (resp. odd), there are
at most (resp. ) possibilities for
the congruence class of modulo . As an application
of these results, we obtain an improvement to the upper bound for the number of
equiangular lines in , that is, we reduce the known upper bound
from to .Comment: 21 pages, fixed typo in Lemma 2.
Modal Decomposition of Feedback Delay Networks
Feedback delay networks (FDNs) belong to a general class of recursive filters
which are widely used in sound synthesis and physical modeling applications. We
present a numerical technique to compute the modal decomposition of the FDN
transfer function. The proposed pole finding algorithm is based on the
Ehrlich-Aberth iteration for matrix polynomials and has improved computational
performance of up to three orders of magnitude compared to a scalar polynomial
root finder. We demonstrate how explicit knowledge of the FDN's modal behavior
facilitates analysis and improvements for artificial reverberation. The
statistical distribution of mode frequency and residue magnitudes demonstrate
that relatively few modes contribute a large portion of impulse response
energy
Developments on spectral characterizations of graphs
AbstractIn [E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241–272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime, some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments
On the Complexity of the Generalized MinRank Problem
We study the complexity of solving the \emph{generalized MinRank problem},
i.e. computing the set of points where the evaluation of a polynomial matrix
has rank at most . A natural algebraic representation of this problem gives
rise to a \emph{determinantal ideal}: the ideal generated by all minors of size
of the matrix. We give new complexity bounds for solving this problem
using Gr\"obner bases algorithms under genericity assumptions on the input
matrix. In particular, these complexity bounds allow us to identify families of
generalized MinRank problems for which the arithmetic complexity of the solving
process is polynomial in the number of solutions. We also provide an algorithm
to compute a rational parametrization of the variety of a 0-dimensional and
radical system of bi-degree . We show that its complexity can be bounded
by using the complexity bounds for the generalized MinRank problem.Comment: 29 page
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