Construction of isodual codes from polycirculant matrices

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over \F_2 in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over \F_2, we show that binary double polycirculant codes are asymptotically good

Asymptotically Good Additive Cyclic Codes Exist

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and ( M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over \F_q, for many classes of $q$'s. This is a partial solution to a fifty year old open problem

Cyclic Diamondoid Structures with Shared Vertices, Edges, or 6-membered Rings

Diamondoid structures with shared vertices, edges, or 6-membered rings can theoretically be curved into toroidal structures whose calculated energy provides information about steric strain. Diamondoid hydrocarbons sharing one vertex between two adamantane units are called [n]spiromantanes, where n indicates the number of adamantane units. When a pair of adamantane units shares one CC bond, the resulting assembly is called one-edge-[n]mantane, specifying (by letters in square brackets) which bonds are shared by the adamantane units. Two adjacent edges may be shared by a pair of adamantane units, and the assembly is called two-edge-[n]mantane, again specifying by letters in square brackets the shared bonds. Catamantanes or perimantanes sharing a 6-membered ring of carbon atoms may form larger rings in an assembly which is called [n]cyclomantane; in the case of catamantanes, the structure of the diamondoid is specified by codes of the dualists. Finally, nanotubes derived from hexagonal diamond, as well as corresponding toroidal structures, are discussed

Multireference approaches for excited states of molecules

Understanding the properties of electronically excited states is a challenging task that becomes increasingly important for numerous applications in chemistry, molecular physics, molecular biology, and materials science. A substantial impact is exerted by the fascinating progress in time-resolved spectroscopy, which leads to a strongly growing demand for theoretical methods to describe the characteristic features of excited states accurately. Whereas for electronic ground state problems of stable molecules the quantum chemical methodology is now so well developed that informed nonexperts can use it efficiently, the situation is entirely different concerning the investigation of excited states. This review is devoted to a specific class of approaches, usually denoted as multireference (MR) methods, the generality of which is needed for solving many spectroscopic or photodynamical problems. However, the understanding and proper application of these MR methods is often found to be difficult due to their complexity and their computational cost. The purpose of this review is to provide an overview of the most important facts about the different theoretical approaches available and to present by means of a collection of characteristic examples useful information, which can guide the reader in performing their own applications

Accelerating the CM method

Given a prime q and a negative discriminant D, the CM method constructs an elliptic curve E/\Fq by obtaining a root of the Hilbert class polynomial H_D(X) modulo q. We consider an approach based on a decomposition of the ring class field defined by H_D, which we adapt to a CRT setting. This yields two algorithms, each of which obtains a root of H_D mod q without necessarily computing any of its coefficients. Heuristically, our approach uses asymptotically less time and space than the standard CM method for almost all D. Under the GRH, and reasonable assumptions about the size of log q relative to |D|, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D), which may be as small as O(|D|^(1/4)log q). The practical efficiency of the algorithms is demonstrated using |D| > 10^16 and q ~ 2^256, and also |D| > 10^15 and q ~ 2^33220. These examples are both an order of magnitude larger than the best previous results obtained with the CM method.Comment: 36 pages, minor edits, to appear in the LMS Journal of Computation and Mathematic

Kinetic model construction using chemoinformatics

Kinetic models of chemical processes not only provide an alternative to costly experiments; they also have the potential to accelerate the pace of innovation in developing new chemical processes or in improving existing ones. Kinetic models are most powerful when they reflect the underlying chemistry by incorporating elementary pathways between individual molecules. The downside of this high level of detail is that the complexity and size of the models also steadily increase, such that the models eventually become too difficult to be manually constructed. Instead, computers are programmed to automate the construction of these models, and make use of graph theory to translate chemical entities such as molecules and reactions into computer-understandable representations. This work studies the use of automated methods to construct kinetic models. More particularly, the need to account for the three-dimensional arrangement of atoms in molecules and reactions of kinetic models is investigated and illustrated by two case studies. First of all, the thermal rearrangement of two monoterpenoids, cis- and trans-2-pinanol, is studied. A kinetic model that accounts for the differences in reactivity and selectivity of both pinanol diastereomers is proposed. Secondly, a kinetic model for the pyrolysis of the fuel “JP-10” is constructed and highlights the use of state-of-the-art techniques for the automated estimation of thermochemistry of polycyclic molecules. A new code is developed for the automated construction of kinetic models and takes advantage of the advances made in the field of chemo-informatics to tackle fundamental issues of previous approaches. Novel algorithms are developed for three important aspects of automated construction of kinetic models: the estimation of symmetry of molecules and reactions, the incorporation of stereochemistry in kinetic models, and the estimation of thermochemical and kinetic data using scalable structure-property methods. Finally, the application of the code is illustrated by the automated construction of a kinetic model for alkylsulfide pyrolysis

A geometric characterization of minimal codes and their asymptotic performance

In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes.Comment: 22 page

Theoretical investigations of surface chemistry in space

In this Thesis, computational models for carbonaceous dust grains were examined and compared to known experimental data. Different formation routes of molecules, important to the astrochemical evolution of the universe, have been investigated and their relative energies were analysed with respect to the harsh conditions in interstellar dark clouds of extremely low pressure (10‐17 bar) and temperature (10 – 20 K). Dust grains are present in the universe, and evidence shows they are siliceous or carbonaceous, possible with an icy mantle surrounding the core. In this research, only carbonaceous surfaces were examined. Two models were used to represent polycyclic, aromatic carbonaceous surfaces: coronene, C24H12, representing a relatively small hydrocarbon, and graphene – a single graphite sheet – which represents an extended carbonaceous surface. The main aims of this Thesis were to examine the validity of computationally modelled astrochemical reactions and to investigate the catalytic effect of dust grain surfaces on these reactions. Several formation reactions were examined, including water, methanol and carbonyl sulfide formation. The abundance of these molecules in dark molecular clouds cannot be explained by solely considering gas phase type reactions, and the influence that the carbonaceous surfaces have on these reactions was investigated in order to examine any catalytic effect that they may have