Standard sequence subgroups in finite fields

This research was partially supported by the Fundacao de Ciencia e Tecnologia, and was undertaken within the "Centro de Estruturas Lineares e Combinatorias da Universidade de Lisboa".In previous work, the authors describe certain configurations which give rise to standard and to non-standard subgroups for linear recurrences of order k=2, while in subsequent work, a number of families of non-standard subgroups for recurrences of order k greater or equal to 2 are described. Here we exhibit two infinite families of standard groups for k greater or equal to 2.preprintpublishe

Intersection density of imprimitive groups of degree pqpq

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on an element of $\Omega$. The \emph{intersection density} of $G$ is the number \rho(G) = \max\left\{ \frac{\mathcal{|F|}}{|G|/|\Omega|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}. Recently, Hujdurovi\'{c} et al. [Finite Fields Appl., 78 (2022), 101975] disproved a conjecture of Meagher et al. (Conjecture~6.6~(3) in [ J.~Combin. Theory, Ser. A 180 (2021), 105390]) by constructing equidistant cyclic codes which yield transitive groups of degree $pq$, where $p = \frac{q^k-1}{q-1}$ and $q$ are odd primes, and whose intersection density equal to $q$. In this paper, we use the cyclic codes given by Hujdurovi\'{c} et al. and their permutation automorphisms to construct a family of transitive groups $G$ of degree $pq$ with $\rho(G) = \frac{q}{k}$, whenever $k<q<p=\frac{q^k-1}{q-1}$ are odd primes. Moreover, we extend their construction using cyclic codes of higher dimension to obtain a new family of transitive groups of degree a product of two odd primes $q<p = \frac{q^k-1}{q-1}$, and whose intersection density are equal to $q$. Finally, we prove that if $G\leq \operatorname{Sym}(\Omega)$ of degree a product of two arbitrary odd primes $p>q$ and $\left\langle \bigcup_{\omega\in \Omega} G_\omega \right\rangle$ is a proper subgroup, then $\rho(G) \in \{1,q\}$

Categorical Quantum Dynamics

We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford University, Michaelmas Term 2016 (273 pages

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum