• Flicker mitigation strategy for a doubly fed induction generator by torque control

      Zhang, Yunqian Q.; Hu, Weihao; Chen, Zhe; Cheng, Ming; Hu, Yanting (IET, 2014-03)
      Owing to the rotational sampling of turbulence, wind shear and tower shadow effects grid connected variable speed wind turbines could lead to the power fluctuations which may produce flicker during continuous operation. A model of an megawatt (MW)-level variable speed wind turbine with a doubly fed induction generator is presented to investigate the flicker mitigation. Taking advantage of the large inertia of the wind turbine rotor, a generator torque control (GTC) strategy is proposed, so that the power oscillation is stored as the kinetic energy of the wind turbine rotor, thus the flicker emission could be reduced. The GTC scheme is proposed and designed according to the generator rotational speed. The simulations are performed on the national renewable energy laboratory 1.5 MW upwind reference wind turbine model. Simulation results show that damping the generator active power by GTC is an effective means for flicker mitigation of variable speed wind turbines during continuous operation. keywords: {asynchronous generators;oscillations;power generation control;torque control;wind power plants;wind turbines;GTC strategy;continuous operation;doubly fed induction generator;flicker emission;flicker mitigation strategy;generator active power;generator torque control;kinetic energy;megawatt-level variable speed wind turbine;power oscillation;tower shadow effects grid connected variable speed wind turbines;turbulence;upwind reference wind turbine model;variable speed wind turbines;wind shear;wind turbine rotor
    • Formal Verification of Astronaut-Rover Teams for Planetary Surface Operations

      Webster, Matt; Dennis, Louise A; Dixon, Clare; Fisher, Michael; Stocker, Richard; Sierhuis, Maarten; University of Liverpool; University of Chester; Ejenta, inc.
      This paper describes an approach to assuring the reliability of autonomous systems for Astronaut-Rover (ASRO) teams using the formal verification of models in the Brahms multi-agent modelling language. Planetary surface rovers have proven essential to several manned and unmanned missions to the moon and Mars. The first rovers were tele- or manuallyoperated, but autonomous systems are increasingly being used to increase the effectiveness and range of rover operations on missions such as the NASA Mars Science Laboratory. It is anticipated that future manned missions to the moon and Mars will use autonomous rovers to assist astronauts during extravehicular activity (EVA), including science, technical and construction operations. These ASRO teams have the potential to significantly increase the safety and efficiency of surface operations. We describe a new Brahms model in which an autonomous rover may perform several different activities including assisting an astronaut during EVA. These activities compete for the autonomous rovers “attention’ and therefore the rover must decide which activity is currently the most important and engage in that activity. The Brahms model also includes an astronaut agent, which models an astronauts predicted behaviour during an EVA. The rover must also respond to the astronauts activities. We show how this Brahms model can be simulated using the Brahms integrated development environment. The model can then also be formally verified with respect to system requirements using the SPIN model checker, through automatic translation from Brahms to PROMELA (the input language for SPIN). We show that such formal verification can be used to determine that mission- and safety critical operations are conducted correctly, and therefore increase the reliability of autonomous systems for planetary rovers in ASRO teams.
    • Fourier spectral methods for some linear stochastic space-fractional partial differential equations

      Liu, Yanmei; Khan, Monzorul; Yan, Yubin; LuLiang University; University of Chester (MDPI, 2016-07-01)
      Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in one-dimensional case are introduced and analyzed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in $L^{2}$- norm are obtained. Numerical examples are given.
    • Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noises

      Liu, Fang; Yan, Yubin; Khan, Monzorul; Lvliang University, University of Chester (EudoxusPress, 2018-02-28)
      Fourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive noises in one-dimensional case are introduced and analyzed. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The space-time noise is approximated by the piecewise constant functions in the time direction and by some appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. For the linear problem, we obtain the precise error estimates in the $L_{2}$ norm and find the relations between the error bounds and the fractional powers. For the nonlinear problem, we introduce the numerical algorithms and MATLAB codes based on the FFT transforms. Our numerical algorithms can be adapted easily to solve other stochastic space fractional partial differential equations with multiplicative noises. Numerical examples for the semilinear stochastic space fractional partial differential equations are given.
    • Fractional boundary value problems: Analysis and numerical methods

      Ford, Neville J.; Morgado, Maria L.; University of Chester ; University of Tras-os-Montes e Alto Douro (Springer, 2011-07-28)
      This journal article discusses nonlinear boundary value problems.
    • Fractional pennes' bioheat equation: Theoretical and numerical studies

      Ferras, Luis L.; Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; Nobrega, Joao M.; University of Minho & University of Chester, University of Chester, UTAD, UNL Lisboa, University of Minho (de Gruyter, 2015-08-04)
      In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bio heat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.
    • A Framework for Web-Based Immersive Analytics

      John, Nigel; Ritsos, Panagiotis; Butcher, Peter W. S. (University of Chester, 2020-08-17)
      The emergence of affordable Virtual Reality (VR) interfaces has reignited the interest of researchers and developers in exploring new, immersive ways to visualise data. In particular, the use of open-standards Web-based technologies for implementing VR experiences in a browser aims to enable their ubiquitous and platform-independent adoption. In addition, such technologies work in synergy with established visualization libraries, through the HTML Document Object Model (DOM). However, creating Immersive Analytics (IA) experiences remains a challenging process, as the systems that are currently available require knowledge of game engines, such as Unity, and are often intrinsically restricted by their development ecosystem. This thesis presents a novel approach to the design, creation and deployment of Immersive Analytics experiences through the use of open-standards Web technologies. It presents <VRIA>, a Web-based framework for creating Immersive Analytics experiences in VR that was developed during this PhD project. <VRIA> is built upon WebXR, A-Frame, React and D3.js, and offers a visualization creation workflow which enables users of different levels of expertise to rapidly develop Immersive Analytics experiences for the Web. The aforementioned reliance on open standards and the synergies with popular visualization libraries make <VRIA> ubiquitous and platform-independent in nature. Moreover, by using WebXR’s progressive enhancement, the experiences <VRIA> is able to create are accessible on a plethora of devices. This thesis presents an elaboration on the motivation for focusing on open-standards Web technologies, presents the <VRIA> visualization creation workflow and details the underlying mechanics of our framework. It reports on optimisation techniques, integrated into <VRIA>, that are necessary for implementing Immersive Analytics experiences with the necessary performance profile on the Web. It discusses scalability implications of the framework and presents a series of use case applications that demonstrate the various features of <VRIA>. Finally, it describes the lessons learned from the development of the framework, discusses current limitations, and outlines further extensions.
    • G-codes over Formal Power Series Rings and Finite Chain Rings

      Dougherty, Steven; Gildea, Joe; Korban, Adrian; University of Scranton; University of Chester (2020-02-29)
      In this work, we define $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$-code is again a $G$-code in this setting. We study the projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$-adic codes over $R_\infty$ to $\gamma$-adic $G$-codes over the same ring. We also study $G$-codes over principal ideal rings.
    • G-Codes, self-dual G-Codes and reversible G-Codes over the Ring Bj,k

      Dougherty, Steven; Gildea, Joe; Korban, Adrian; Sahinkaya, Serap; Tarsus University; University of Chester (Springer, 2021-05-03)
      In this work, we study a new family of rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj,k to a code over Bl,m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s.
    • Galerkin finite element approximation of a stochastic semilinear fractional subdiffusion with fractionally integrated additive noise

      Yan, Yubin; Kang, Wenyan; Egwu, Bernard; Pani, Amiya; University of Chester, Lvliang University, P. R. China, Indian Institute of Technology Bombay
      A Galerkin finite element method is applied to approximate the solution of a semilinear stochastic space and time fractional subdiffusion problem with the Caputo fractional derivative of the order $ \alpha \in (0, 1)$, driven by fractionally integrated additive noise. After discussing the existence, uniqueness and regularity results, we approximate the noise with the piecewise constant function in time in order to obtain a regularized stochastic fractional subdiffusion problem. The regularized problem is then approximated by using the finite element method in spatial direction. The mean squared errors are proved based on the sharp estimates of the various Mittag-Leffler functions involved in the integrals. Numerical experiments are conducted to show that the numerical results are consistent with the theoretical findings.
    • Galerkin methods for a Schroedinger-type equation with a dynamical boundary condition in two dimensions

      Antonopoulou, Dimitra; University of Chester (EDP Sciences / SMAI, 2015-06-30)
      In this paper, we consider a two-dimensional Schodinger-type equation with a dynamical boundary condition. This model describes the long-range sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a Crank-Nicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution.
    • Gastrointestinal Stents: Materials and Designs

      Black, Steven J.; Edwards, Derek W.; Smith, Graham C.; Laasch, Hans-Ulrich; MDECON Ltd.; The Christie NHS Foundation Trust; University of Chester (Thieme Publishing, 2018-05-09)
      Over the last 25 years stents have developed into an established way of restoring luminal patency throughout the gastrointestinal tract. Materials used as well as the construction of these devices have become more and more sophisticated in order to comply better with the complex environment they are inserted. The requirements vary greatly from organ to organ and stent behavior differs significantly between stent constructions. However this is not necessarily understood by many operators, as the choice of devices is now vast and in many cases decisions are made on availability and cost. An increasing challenge in malignant conditions is the improving survival of incurable patients, which now exceeds the traditional life expectancy of a stent by a factor of 2 to 3. Consequently more patients experience failure of their stent and require repeat interventions. This has a poor impact on patients’ quality of life and potentially on their survival. Re-intervention is often more difficult, carries the risk of additional complications and presents an additional economic burden to the health systems. This article illustrates current stent designs, their different behavior and their limitations.
    • A genetic-algorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombination

      Bocharov, Gennady; Ford, Neville J.; Edwards, John T.; Breinig, Tanja; Wain-Hobson, Simon; Meyerhans, Andreas; Institute of Numerical Mathematics, Russian Academy of Sciences ; University of Chester ; University of Chester ; University of the Saarland ; Unité de Rétrovirologie Moléculaire, Institut Pasteur ; University of the Saarland (Society for General Microbiology / High Wire Press, 2005-11-01)
      It has been previously shown that the majority of human immunodeficiency virus type 1 (HIV-1)-infected splenocytes can harbour multiple, divergent proviruses with a copy number ranging from one to eight. This implies that, besides point mutations, recombination should be considered as an important mechanism in the evolution of HIV within an infected host. To explore in detail the possible contributions of multi-infection and recombination to HIV evolution, the effects of major microscopic parameters of HIV replication (i.e. the point-mutation rate, the crossover number, the recombination rate and the provirus copy number) on macroscopic characteristics (such as the Hamming distance and the abundance of n-point mutants) have been simulated in silico. Simulations predict that multiple provirus copies per infected cell and recombination act in synergy to speed up the development of sequence diversity. Point mutations can be fixed for some time without fitness selection. The time needed for the selection of multiple mutations with increased fitness is highly variable, supporting the view that stochastic processes may contribute substantially to the kinetics of HIV variation in vivo.
    • Gradient-based optimization method for producing a contoured beam with single-fed reflector antenna

      Lian, Peiyuan; Wang, Congsi; Xiang, Binbin; Shi, Yu; Xue, Song; Xidian University; University of Chester; Chinese Academy of Sciences (IEEE, 2019-03-07)
      A gradient-based optimization method for producing a contoured beam by using a single-fed reflector antenna is presented. First, a quick and accurate pattern approximation formula based on physical optics (PO) is adopted to calculate the gradients of the directivity with respect to reflector's nodal displacements. Because the approximation formula is a linear function of nodal displacements, the gradient can be easily derived. Then, the method of the steepest descent is adopted, and an optimization iteration procedure is proposed. The iteration procedure includes two loops: an inner loop and an outer loop. In the inner loop, the gradient and pattern are calculated by matrix operation, which is very fast by using the pre-calculated data in the outer loop. In the outer loop, the ideal terms used in the inner loop to calculate the gradient and pattern are updated, and the real pattern is calculated by the PO method. Due to the high approximation accuracy, when the outer loop is performed once, the inner loop can be performed many times, which will save much time because the integration is replaced by matrix operation. In the end, a contoured beam covering the continental United States (CONUS) is designed, and simulation results show the effectiveness of the proposed algorithm.
    • Graphene Oxide Bulk Modified Screen-Printed Electrodes Provide Beneficial Electroanalytical Sensing Capabilities

      Rowley-Neale, Samuel; Brownson, Dale; Smith, Graham; Banks, Craig; Manchester Metropolitan University; University of Chester (MDPI, 2020-03-19)
      We demonstrate a facile methodology for the mass production of graphene oxide (GO) bulk modified screen-printed electrodes (GO-SPEs) that are economical, highly reproducible and provide analytically useful outputs. Through fabricating GO-SPEs with varying percentage mass incorporations (2.5, 5, 7.5 and 10%) of GO, an electrocatalytic effect towards the chosen electroanalytical probes is observed, that increases with greater GO incorporated compared to bare/ graphite SPEs. The optimum mass ratio of 10% GO to 90% carbon ink displays an electroanalytical signal towards dopamine (DA) and uric acid (UA), which is ca. ×10 greater in magnitude than that achievable at a bare/unmodified graphite SPE. Furthermore, 10% GO-SPEs exhibit a competitively low limit of detection (3σ) towards DA at ca. 81 nM, which is superior to that of a bare/unmodified graphite SPE at ca. 780 nM. The improved analytical response is attributed to the large number of oxygenated species inhabiting the edge and defect sites of the GO nanosheets, which are available to exhibit electrocatalytic responses towards inner-sphere electrochemical analytes. Our reported methodology is simple, scalable, and cost effective for the fabrication of GO-SPEs, that display highly competitive LODs, and is of significant interest for use in commercial and medicinal applications
    • Graphene oxide electrochemistry: the electrochemistry of graphene oxide modified electrodes reveals coverage dependent beneficial electrocatalysis

      Brownson, Dale A. C.; Smith, Graham C.; Banks, Craig E.; University of Chester; Manchester Metropolitan University (The Royal Society, 2017-11-15)
      The modification of electrode surfaces is widely implemented in order to try and improve electron transfer kinetics and surface interactions, most recently using graphene related materials. Currently, the use of ‘as is’ graphene oxide (GO) has been largely overlooked, with the vast majority of researchers choosing to reduce GO to graphene or use it as part of a composite electrode. In this paper, ‘as is’ GO is explored and electrochemically characterized using a range of electrochemical redox probes, namely potassium ferrocyanide(II), N,N,N ,N -tetramethyl-p-phenylenediamine (TMPD), dopamine hydrochloride and epinephrine. Furthermore, the electroanalytical efficacy of GO is explored towards the sensing of dopamine hydrochloride and epinephrine via cyclic voltammetry. The electrochemical response of GO is benchmarked against pristine graphene and edge plane-/basal plane pyrolytic graphite (EPPG and BPPG respectively) alternatives, where the GO shows an enhanced electrochemical/electroanalytical response. When using GO as an electrode material, the electrochemical response of the analytes studied herein deviate from that expected and exhibit altered electrochemical responses. The oxygenated species encompassing GO strongly influence and dominate the observed voltammetry, which is crucially coverage dependent. GO electrocatalysis is observed, which is attributed to the presence of beneficial oxygenated species dictating the response in specific cases, demonstrating potential for advantageous electroanalysis to be realized. Note however, that crucial coverage based regions are observed at GO modified electrodes, owing to the synergy of edge plane sites and oxygenated species. We report the true beneficial electrochemistry of GO, which has enormous potential to be beneficially used in various electrochemical applications ‘as is’ rather than be simply used as a precursor to making graphene and is truly a fascinating member of the graphene family
    • Graphite Felt: A New Material for Electroanalysis?

      Davies, Trevor J.; University of Chester (2015)
      Limit of detection is a key property of any sensor. For electrochemical sensors, a common and successful route to decreasing the limit of detection is maximising current density, thus boosting the signal to noise ratio. In quiescent solutions this is achieved by using micro and nano sized electrodes, where decreasing the electrode size increases the mass transport coefficient.1-2 However, as the electrode size decreases the fabrication technique becomes more complicate and the cost of the electrode often increases. In addition, the magnitude of the current decreases, eventually requiring the need for high specification potentiostats. This presentation will introduce a promising new type of electrode for electroanalysis based on graphite felt – a commonly used electrode material in redox flow batteries.3 The electrode is porous with a large specific surface area, is easy to fabricate (Figure 1) and has an approximate cost of 1 pence (not including the platinum wire, that can be re-used hundreds of times).4 Surprisingly, low limits of detection are possible with this electrode, typically 10-100 times lower than conventional carbon macroelectrodes. The reasons for this will be explored, along with an explanation of the distinctive voltammetry observed with graphite felt electrodes. Given the low cost, low limit of detection and relatively high currents, graphite felt is a promising material for electroanalysis that warrants further investigation. References: [1] Henstridge, M.C.; Compton, R.G. The Chemical Record, 2012, 12, 63 [2] Dawson, K.; Wahl, A.; O’Riordan, A. J Phys. Chem. C, 2012, 116, 14665. [3] Chakrabarti, M.H.; Brandon, N.P.; Hajimolana, S.A.; Tariq, F.; Yufit, V.; Hashim, M.A.; Hussain, M.A.; Low, C.T.J.; Aravind, P.V. J. Power Sources, 2014, 253, 150. [4] Smith, R.E.G.; Davies, T.J.; Baynes, N.B.; Nichols, R.J. J. Electroanal. Chem. 2015, 747, 29.
    • Graphite Screen-Printed Electrodes Applied for the Accurate and Reagentless Sensing of pH

      Galdino, Flávia E.; Smith, Jamie P.; Kwamou, Sophie I.; Kampouris, Dimitrios K.; Iniesta, Jesus; Smith, Graham C.; Bonacin, Juliano A.; Banks, Craig E.; University of Campinas, Brazil (Galdino, Bonacin), Manchester Metropolitain University (JP Smith, Kampouris, Banks), Universite Paris Est Creteil, France (Kwamou), University of Alicante, Spain (Inestia), University of Chester (GC Smith) (American Chemical Society, 2015-11-12)
      A reagentless pH sensor based upon disposable and economical graphite screen-printed electrodes (GSPEs) is demonstrated for the first time. The voltammetric pH sensor utilises GSPEs which are chemically pre-treated to form surface immobilised oxygenated species that when their redox behaviour is monitored, give a Nernstian response over a large pH range (1-13). An excellent experimental correlation is observed between the voltammetric potential and pH over the entire pH range of 1-13, such a response is not usually expected but rather deviation from linearity is encountered at alkaline pH values; absence of this has previously been attributed to a change in pKa value of surface immobilised groups. This non-deviation, which is observed here in the case of our facile produced reagentless pH sensor and also reported in the literature for pH sensitive compounds immobilized upon carbon electrodes/surfaces,where a linear response is observed over the entire pH range, is explained alternatively for the first time. The performance of the GSPE pH sensor is directly compared with a glass pH probe and applied to the measurement of pH in real samples where an excellent correlation between the two protocols is observed validating the proposed GSPE pH sensor.
    • Group Codes, Composite Group Codes and Constructions of Self-Dual Codes

      Gildea, Joe; Korban, Adrian (University of Chester, 2021-08)
      The main research presented in this thesis is around constructing binary self-dual codes using group rings together with some well-known code construction methods and the study of group codes and composite group codes over different alphabets. Both these families of codes are generated by the elements that come from group rings. A search for binary self-dual codes with new weight enumerators is an ongoing research area in algebraic coding theory. For this reason, we present a generator matrix in which we employ the idea of a bisymmetric matrix with its entries being the block matrices that come from group rings and give the necessary conditions for this generator matrix to produce a self-dual code over a fi nite commutative Frobenius ring. Together with our generator matrix and some well-known code construction methods, we find many binary self-dual codes with parameters [68, 34, 12] that have weight enumerators that were not known in the literature before. There is an extensive literature on the study of different families of codes over different alphabets and speci fically finite fi elds and finite commutative rings. The study of codes over rings opens up a new direction for constructing new binary self-dual codes with a rich automorphism group via the algebraic structure of the rings through the Gray maps associated with them. In this thesis, we introduce a new family of rings, study its algebraic structure and show that each member of this family is a commutative Frobenius ring. Moreover, we study group codes over this new family of rings and show that one can obtain codes with a rich automorphism group via the associated Gray map. We extend a well established isomorphism between group rings and the subring of the n x n matrices and show its applications to algebraic coding theory. Our extension enables one to construct many complex n x n matrices over the ring R that are fully de ned by the elements appearing in the first row. This property allows one to build generator matrices with these complex matrices so that the search field is practical in terms of the computational times. We show how these complex matrices are constructed using group rings, study their properties and present many interesting examples of complex matrices over the ring R. Using our extended isomorphism, we de ne a new family of codes which we call the composite group codes or for simplicity, composite G-codes. We show that these new codes are ideals in the group ring RG and prove that the dual of a composite G-code is also a composite G-code. Moreover, we study generator matrices of the form [In | Ω(v)]; where In is the n x n identity matrix and Ω(v) is the composite matrix that comes from the extended isomorphism mentioned earlier. In particular, we show when such generator matrices produce self-dual codes over finite commutative Frobenius rings. Additionally, together with some generator matrices of the type [In | Ω(v)] and the well-known extension and neighbour methods, we fi nd many new binary self-dual codes with parameters [68, 34, 12]. Lastly in this work, we study composite G-codes over formal power series rings and finite chain rings. We extend many known results on projections and lifts of codes over these alphabets. We also extend some known results on γadic codes over the infi nite ring R∞
    • Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes

      Dougherty, Steven; Gildea, Joe; Taylor, Rhian; Tylyshchak, Alexander; University of Scranton; University of Chester; Uzhgorod State University (Springer, 2017-11-15)
      We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in [13] by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes.