19,215 research outputs found
A general construction of Ordered Orthogonal Arrays using LFSRs
In \cite{Castoldi}, q^t \by (q+1)t ordered orthogonal arrays (OOAs) of
strength over the alphabet \FF_q were constructed using linear feedback
shift register sequences (LFSRs) defined by {\em primitive} polynomials in
\FF_q[x]. In this paper we extend this result to all polynomials in
\FF_q[x] which satisfy some fairly simple restrictions, restrictions that are
automatically satisfied by primitive polynomials. While these restrictions
sometimes reduce the number of columns produced from to a smaller
multiple of , in many cases we still obtain the maximum number of columns in
the constructed OOA when using non-primitive polynomials. For small values of
and , we generate OOAs in this manner for all permissible polynomials of
degree in \FF_q[x] and compare the results to the ones produced in
\cite{Castoldi}, \cite{Rosenbloom} and \cite{Skriganov} showing how close the
arrays are to being "full" orthogonal arrays. Unusually for finite fields, our
arrays based on non-primitive irreducible and even reducible polynomials are
closer to orthogonal arrays than those built from primitive polynomials
Free reflection multiarrangements and quasi-invariants
To a complex reflection arrangement with an invariant multiplicity function
one can relate the space of logarithmic vector fields and the space of
quasi-invariants, which are both modules over invariant polynomials. We
establish a close relation between these modules. Berest-Chalykh freeness
results for the module of quasi-invariants lead to new free complex reflection
multiarrangements. K. Saito's primitive derivative gives a linear map between
certain spaces of quasi-invariants.
We also establish a close relation between non-homogeneous quasi-invariants
for root systems and logarithmic vector fields for the extended Catalan
arrangements. As an application, we prove the freeness of Catalan arrangements
corresponding to the non-reduced root system .Comment: 26 pages; small change
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